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</style></head><body><div class = "content"><div class = 'SectionBlock containment'><h1 class = "S1"><span class = "S2">Exercise 1: Classification of partial differential equations</span></h1><div class = "S3"><span class = "S2">Date of exercise: 8th May 2019</span></div><div class = "S3"><span class = "S2">Last update: 16th May 2019 by Lukas Babor</span></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><h1 class = "S1"><span class = "S2">1.1 Classification of PDEs</span></h1><div class = "S3"><span class = "S2">Consider an example of a partial differential equation</span></div><div class = "S4"><span style="vertical-align:-18px"><img 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+7gGurW0u2nePyToxvVqw2urgWrVebGOIZIziXWyORWlL4YFhMfIN1/9JG80r74AmphXOj4ltP4xl5TjtwwW8LR0lmxttTpdvvDGB8I2m0dHEC7Y1ME4ULSER1G9Xn/HEY4BJmVo2eH1n4m93R+AFNZM4GMTi37I9uLwAvwVEfVE9isucbdsdlzXSsOuxVP89YkM3MeJkY8hkFmYorpSf0sI5VWEd4l91oDZ3jqWwIYL7m/tX6DmYyxnSpFeFZKKKL7MbKTur1N3ZsjMBTgOTVRUalsjI23BVisrptC7CcIcy1dbXFeZY/hqONTTtMW9zR2DTgBmx4OCgBYslWHCTY/xpZMfavKPjZhMkZGle1t9tGaX6Req86PfJ9lE3WRdLlvDcgU68Jh1LX74j6Q3ZRZ+w0FCKxs/ERJZ4lxd3xyetcq3cpDXuMxQl1rqZr6WLRRaWPhMavjbtWaU9f5IbEfhagC8xYGLilMiOmhQLdDuU9tpPqMs2VwCvyY13AO9flW5ECiMm1MIis9zIzg0cWaoLMZrVqgwYddDHtYa36v7mXurIBr18GpUKpjlJWMbMa12HNlkzjsU+1rWYLT+knn/l2LzVyrZvbPq9HDeHS5Xw14b2SNZcoYdCWdmqwraa+ffM+XGnfRro33TSudin681LnpHkndObRBB4B+VGtJEYlJQ/rYTjFGrbHe9NKXcJ6vcV9rn1M0baMDcvTS1oAhVLzePc9litRHmSh7smWV9GgnSNK7r66JGCnxIxwBEV4rSWY8DgSn8B4E05qOAvpP0a0gmlgyMXIN3eQMiFNgyjWumxvt2HspwgJARnXMYU1WxPo+kgHTnw1BuzplZKn0vWdXh73KwKIu96uBusyHBAmYshadQNkW/MSfLzXuzuHFssnqyUzhUfjYkx70SCrXFy3dBbimM4CBZ/SlQCjQ3yeX81brypowQ1c7Ultd/5bXkdJmdHmtWuNkS2xwiGw8Fyl76fepUlCm9q3BRpp929qPpD+1P2YaVU+xGikHoAUjGiOoJ/wWA4bWfGZpoOprANnq6V9rYuSc9SRmY2vqXHVRBljdHFjT4BclJGA72U4ZHWVNWJIxgRdwjP0ZnkFWhOPEwZVZ1AJng89hMiiORfcCqqlzztFfP2Y/Gqi7K0619D55xy2Mu5PVNaQ8uZfMxrynj/3SR6XOC+iSCG0saQwAo28T21rSkweY2HJQzdbrnLoej1S5R6SCZEZJRKSiBgyNXGdYhucASJKSlzr2Uch+iObVPKVFtRINvWCZGxmBpWKD9g0sGUSz9B7WOl5dTzq4lj11cqBi/Zpyc7cgeLG3QsvD5shwykIJgHRzeB9hRU4JAKM6uiAJ66LxwwCN3nllQan/YRfGEccMr/V+3tt5kzfj4r2plXGnyJosJBU7BWI6iuJ525DOMhG3RARnBAgln33frXQoypD6w96mnsMAaiGDPpd5MZEqwT+trjn3PN5XGxp06BFIk0TsWkRPDayZqwvHKSG4howGB3BL8Y4oU9V85qJHGZE9jGitwZQiaA6WyJkBlG3EQun9FtXL0OX8gTV0qNsFVkZ54EjBjiyDiYvKYt+2gmJk3nB3CFiAxZQ8Od8yakh9ZOxqTn1OGma/Wk7Ni8Aemq2xsk2yfUeS3D9O3bNMpm9xiMtokpbPC/tYtMdJHaVIb3pwUqpd5mtAe6E+/FYEK+6EobTZ2dgYdNGAbEvORPfREdOqfORdrtdAossZjjV63Rkq3BJbkinVWpBJ8W1zEFPgIkSbW06MHHsyCZSU1YG9kOExEZCjbEN0j/lMGHl4DNFxl1drAGZACQBwr6Vxt9+YESKr5VlwsIyvBj7FnfAO5b89O8EFzBHRAqYsoSznN+TUvIiPKuPC/e6IXUYyN7LH9n7BF5FBrnEa9uuk136Nd7dl5O7EIj9euzRSig+Gpzw2Q7t/SJdpIOc+jzGwXXaG1/wVR0xNUJuQc+v/hSnI9LYPSOcAseklf/Wcaz3yMTmMRqZBi11rubVxtx6gRKQrtYiWRM63FI4qDICsG8Df5xQcTbY52qN8T2rX5BSXwi3Xcctrr2HclbtkUThWQK6i66FTVdNOj/EAJEEU3ZyaF+GZcwJkXbW8HrFF2xp1DCXOC/j3omC5tXF3I1DB9eQtLRvaPrZqG7vlx7DWtRMY1KKf1bxz3p/+WpkWL/8Q+Jh1aKkeRqk7ZvOerNqqmeSn+pXnnfk7j9qDcbeH2EQzsnBfLTNMc/TjnnwzUuFkqg1SxkFbkdIb0RefnO5zrntvx7Q27kim7DlS8LglZFSmMgJwMf5+LCuVVNz0fQpclkGR8hFg61FFds63rqRjP76oJLmGcXdOhB3IHIA+GPbk/j6qcm993xwjoBxpppwkNeeepL2wd3k2XnwRun8PJefGj1Fjzrnuox6TLVHWL1J8Sl1soWL2YNwtwabNiReNAZxKt94SowJwmdMcvcPYJ6d0nxundZyii134qA55eGvjroaOe4KcY9rMerO0tJbSWpIH4FSLoyAmWSGVZDzbawSIVzA1V2QnDP1pIXAHWlCvwu6sYdx96NK4PCr19rXmrrdQ2hHO4RklCGcpGjkny1m3dJjBLGOSXjNENAKdLvM0UKespQUv8rivuNRejDtWSi2G0qvKOLeut6deqBIg7BQg9BMKrsRvl35fVzy6Q/y0pXGv6b1hiLB9nupyQSbFhgy7k2oqbt0o2Q44VCgQZY53xqqo5fUa0XLMYZ0rrbJX+CRwIgjMlE5xrFwkrY07tKOP3AfOQ5syHhfdXP/RExqojTtazSFa9FI1+Sikwni0p8aaeoZqxFKWjNPzL71A/91TNADLkN7/vRp3YB/OvM4SmJuMmG/5OtTDnWQTaiav4X2Z3JYOrKyY7FiX79RAS+NeO1FLxiDX8yKA7ab4KOry8BTH/xGeM8f5C8qNWtPFHCutjXuOuTyFYDyCQvd2j9rSGN0lH4UIJtNEp6IZCOjMtgznTNMLsJQPDPL42kiUY6HFhQFA7nHvUqflGb85bVDYHqe8dZ0pADb4D4apy9Qt46UUs4aoaQP3aIdDvOI+9iBSom9d2qjO8RWggvU+kyd5uUupwbuqV/6e31UO6JBOOJ+hkqoSh29+qsxir5969+p3ITt+/Bkg71yyHSNws3vnFFNolltkBwCIBSxHFN1KHCNy1YyKlsZdRKkPcU5944hK3/KegU68oFDJkJNzRLsJQ0FOTSnT+sJjJ8MWGP2U+lBFQjb0a4F2QFe4trdmDJujrxbHSPPKttgYOWdzpI7+5xz/3GpzmHP81DEyO9KeEOl7mu1uT/O9AB4JNoZEO/V68L/7rawWkQrOzEO+q9jhECDdq9T7wpw1jg2+GZ7HviG1nLJkPkbigZyL41b3wdfXRXIlAOFUcFCOKjUznZK38b4XSSvjnoNNpOTVQPbQcpAb60WK2PBHakiiimsk2ajUjYZTuS45j01MHVT95hzDUQJZvm5AUJLX8TG7n1Opzql78r6JkqBXZSPGRLS0Rxa+sSzGOd3nOFqDfhK5fe6Ya/7+ljX3TykgpbFWpWvW0Pq3DHT2AQs4OFdT4jsU5ROMcUoMKfmuwp9gRTwl9Xlar2fp+Zbs8a3S8oxsct8vnT2vfU4vvP3DfjaWDfOt2e+sfWxa5FKdbnl84p/wRwD5XSxLHnxeRD1Da5WPw8bdkuzg4oWM/PCejDvdSsHaUN97hlLqUbWnIhFOgLY4lK23+ijuybgniQ1nCDiotdzKuGc9E8AHA98pYFTrNZ87n1RtTkg7B6aro8AlFLXduJ9+KvUApyWRNFAZngKZli8rRERjV66nQc7FUTCko+Qw517A8vf2Mfv1EpEds1778tUZpKXGndFQs/JxM+weXJf2GkjvbWqu+bkr8nR5vEQtyoY8JtKrmKNITR1cIzb93bUbobaXGlwJiaqd5d4l+dfPZUvm6uEWxl2XhZYmIDop+akMzNw1LT3ODAIkJ+TUqN03KPV05RJRnogf2U09Rtg5jgzGWqrLVpE7Xv1MJ8v4wd5MiRQ6wy0i1/5VTxqs29u0W6tBj0nOdPB3Y2N3L9HLHtDysBCynORcFuopa1pi3EV3PnB9iAASpml1WUcDWRejY5vNtVJ/XF86AaCpCSacv2Z+4jF73iI2NeBr53fr0YcWB7iSgRjlQr52UQf4vSldjMVa7GdbG3cjSkUkMhGc+TX52Oc+3jrzMzWLAecA7AcQIKl5xxl776chLdYoxVvXi+fe1xGPa2Xcrd3ERe8PYcDHAkH7jAgVsRZabARQtdij8p2bKm86FqlSyutPcHkc4Xm8e8GwuderKabnGndpAghctVGAlUyDHUFhR7zHF5V0/JIIEB5C3ZgMIxrAFAMYfAiMOBkD3dkszT83Je9aYiKbphSTLAQU7iNIbmgGVjAgrWVL4+6ZWw9HTavrHueH28/gGxhn94qVzMRKHQP1Zu97Sm7yKZCqaF6Jsa7Dt35+ez9fS+NeG137CHpis0dSACCh4NkTtsX88iF2q2Ye1KkzjOwFmcbfev6CzrVmOmz13GCsTBUkgqKrZh/MNe42q3c7Qeu45uKlSwwO4OE9iiR+wIYjCp8jSCREzlD3RB3fGEHnk7Y3/tTLI6Xpo3ph+Z2e4WR64m0z/lD010qWBrTjobd9BAGC4TAtZRac0tVWxt3Gq9cWMEpGB8ZmS8E1jlDE+3pKbPaJnlZrVU7SqqWdDQreu6vVU1uiDga/GWMhU2/lvJq0x5l9VGlp3Onw/cvoZhE65kB7uI4bDpR2Q1mSlxS+/6lyj4g+WTql+mU1BQ1S2PYl9XatvJ499k1lSKKdDM/HkUSmVqurDqUMui6+/znGnWIp+FybycU3ceUPeX24onEUXypSwuqDMAJeIJsulOy5zeLS86/9O9FHsmwtaSPRvibVxcOV4uIsGBCj3cJGlmnkXA/POse/+reoyGCgOWj2XAPDfiktIxCa58ZrlUqF4JdOVRvdu8hu2aB8Y3pzs72w5X1vYdzp3QaK8UubpKh3S7FJ2+SkdM914cCS6PYgSW8KR+K+GX7z1PVka+EDdpzCCIkaGZ6r65zlPUU8wpmW/RI9ylZxjvYEPLzkGbY27q7JCOv4AUJMkKlnIvCwx8m4nBKOAaZNmTDpfU4ah449sq/YK8aySqfafy/Rxda/sW8oB7Fdsq4ow6+Sa427zYQSEVe42LkWkatu5oIfa7kTWUJfXir40xmk7DNGZuC+zSiW/mFsRLF7lreoZmSLPDDMbS08SJvs3Ha8TyweuZR8Tpw7tQalAym2FEAcKFlGEwpV/T8nfG2ti0uuV3co+FZ8oEeUTyrfyBKOhSXr5nja5GzoWwkwqRY/GQMZrUtFhgOmJcdeMzwcUYZFIMHJy9a7S8/Zf/eYGqhbOmd9e9cYd3U2GxT0IqDJxc30jZ4NT1zrDUBXpowvPbVo1VqVE6SeedA8eUoThYpKoP73KjYNUTN5VkErb32vRmMy0Frx5rR4cAhFXzIAl0wI5IyJEjkUGLLUPz1DHj+DYxAIDMEeQV2eDSQv1C6RUpNaO5rkaE1dGgzdHIa9uWv2rFFt+t7NXoeg3kpywAkHTUr/GgE2hdAWhUI6e08FEtaBgIrTmoxj15y3//axNFCD6d6pZM+u0sClxl2NW7pE3x0jP8UKdNXFL/yxNLSas1SONLFxga0cCykeqFhRqYh4z6186kVATde2oV2o5rM/yxQdA33tMCB6Fn1ZA8zEUlFSkL1g2KVAbzmBbGotySxnzb6fowkjpQTGQL3lDDbCJetlVGWION23GNdpf4OV4FS3TKV/UETgQz81tGmJ3vqx96OB7JCyx8GAXL3HXWLckz9afUNbz1bpRT2Namj4uW3mRIpMBNFS3qT0XEvRW1/Lj7nlfXJCeHBze92X3ovMhhLHVOQtZc5BEqEPx2smQ1XLKXPJMe39WEI2sVQvU8dnj/tUO9Za121xXihzNVDZBs4cYqMtxHuCeAaftvILmTPRa8m9qssr3bgyYw4AABCDSURBVAlmOI4tBX4BUA+gz1qPmM1pqY9+rnENKOPp42efYZWUpq6Wc8ZdpA5k5WKzL3LhXUE6MrQ2EylNoC8fQy2z0hMXXJ9BkpHYM2FFAhmVRRBwbCkZedvkGXhp9SHeItv1tLPYnGtJ1rCWU+Y4ne6HRyubc+0Qm7X19+UR8fSC7D3SpDHvFsMmaoX1UENeQ+w9AFWAZ3AY0NJDLE0+2y0BlPYf60fmpM7P0E/Ngp+jl6QkvlUGbs4992O20YCMjoyZ/ZID6F3JzoCr7+CccU96Rh+XNEELcdO8cjVVBl0rlhTYOe5txBnSE3VvZIv7cY5MoULaPrvVSRufpx51yJhtFU1ZRg0O8/9j3AaAiTAMwIn452tRO1VvhHxvCSjK1NVcHEDjR/Tk6XLAjned04js6QgCmQt8CtGMQTBZDZfeuyDBP6Ji+APfvG85B7ZMnf8WFMjepbojoDUJSk5dhBAHvNsSx7D0Ofbj19WAb0/pFc4ILkPQOVtOGfd6Hu7sCzQ8cBZi8MLrI1RA4gKRbf6z1LxoldduQ0o5B2bTM8vbxybUug+foeBk6VPXy2nj25sw3tZuElktSjmyIsorLWcPyBJIzwPceYakdoL8vwwD52RqLCVSFiUZv5GGbdE5AVDF8fC8lJeOsIF7v7RD5uTAPbxbWtJqtrE93NPSe6gpVGV29HsT7XkwBilAo0CsUyLCU6ZEt2uv7tI18BoamDLuSAWQRswdU9lazYxtRhOtz+18datZImRlFfTzM6YZzQPaAISNpekQNHzcAjT5JevyMTOecwfIXHKNub9BaqRWqw8ddiH5vUVqsgwoWJMVbO41hseNlSqkVd2LEk8y8slAqeUOhQPHSVIC0jWRPM5L748jqgRxisd/6TVaHy8bg0CETvYgMnWimJYp8T2si345nEqOdcYJaBiqXv+23n4lCR09Y8QrnFBEQt5dffx7xQntQd8Pew/n0vKPopi6p3AsdQxMJiIkNQ916if7/32giC/WkkwZikLRK+5JpHT1nWs7fI9iLJG4iEDodA3KUgYbcYXNTamnLtnYKEXh7suf52znWmci//cs0WpL1jVOAmeUk/HSPT2kfi+70ADmNHiDMZIc4NAPLnep7q8lqhZU0dK2OlZgkM4R++xiwf0mttfAoxl3KXabrg8Li5F/GIZ6JKH0rjRvLSI76EXCYPCo86NiuIAORT1r06rKJoiCZVT2OllNKpWBFzlrf1NH8v85Y3vOW85Qo6+1bpta4g2UQJ5fTijdOZwxj4gEIQkZ4ilE/TZXWZmWsxG0vWHS49jARnRE9JwnfvxjROYMsb2CMUfclM4nzorkUh+CT2W67DH+XtZC6j6/He+TQMN7ju9hzyROx3+CB1/Boxh3H9qHFZ5ptcUUlJCiPwxS0sYkaSuHj9YUIv2+RC1MRMpR8OdAYlr0tkghSsW51tjUpIO/jk+5fUZdCQKSOoUzJqIBglQKgTomY0QxnCEbpdKKujdkNqCYzIL2EudQs2wpgIMAhC3b/lreXz/X+hqAiZFRqmcwyLbZNwQCaeSRcSG1GQp67ZyIhlAH1sc7zFEWoEjL75W8aX3t9itcpIFHMO7StcAqWX99cWkxMAxFC440mOgqZwP7nbGRQ2HYGXLCExfNcwhEaUZHXsK6dtFDOfMjQBrAJ0YLV/y91tvUWzlf0MqcJgQgGAY5Z4BHUpfa37Tm0QEHbgy4ZpO0WRI954w6rgAbq7p4a8lJcGr4esW7PJYG6sEmomwOKHY9mSzAN8ZZdot4L5XyhsKQ+745B95tJUEOo0wAwq1rWfMe6wn01T6hgUcw7tmGZb0fU0hxaoOoHQdBDGNNtHHpUR4Tqfn8HWMO7AKMZwDLVqKlyIfP6O25L3+JPoCObIzAcaIc2ZRh94FWKqlOWReodFH+mEhv0ldOVfLcGHcAstaOkcyAvnBpVDX+Lo+lgeySsGodIhzxmn/BfqsUlL3LukqmqGiT3965vLNJt9yqPfGxnswDrvaUcedx8iwhn+fwrouKRE1zj8/HAYEt2sYvrVYORXupqFHZ2EXvBjfwnsdS54lCV0cHwJqi+hu2WdWtLGP3xPg6N6eiZfo3kfl77su/9BmN/U7ZIVv9pjoDvLuMuuhGRkUNckpE+clihwRIN8gldI7q+LI4xoXWpYGp6+SQFal5iPl7ES2M9CBivEQPl6x7yXd9yfm3/o33kWMnOOCQ6roZQ7pj3DORjkxlCf2dfcjelU6r2jwA6JSs8Yy21mG/XkMNnDLu0otATOo7l0zxGt6W4xE2QEnPOV7/sTq5l1q06v+vNe484xwyo84K9DYmOOVt+D5OdfQpsdlzEnImOg/8FPsYHmnp5JbMbO5Nas8AHMQ/7nuqh7vhq7LpqV5RonUXRXs8xvmfM7f9xmbJ4ZmSeoiL38i2SPmfE7PLtfbpWYdMPiUyBEYJA9Ipl+BIuBdJPcieiEaXSIvvesn11zqWUc/hNp9WWtTGrqUsZC8ipwYKKT9xqLyrRDeH8t+UtHxGa+mon3dDDZwy7mgoGZG5M8+ljXmfc1HSUuDQ0VqV/Lca7LXG3UdmTCyx4aqPDwWbGFQ3GaNOzd9bC+Cc++BN051aPXR0TmwbntsxsgVznJtzrwFOdY4FtPgateNz11/z7/FvM96nWv7qFOhYh0PeH/1rHRKpIyginqMuh3PiGcvWaG1jtE9J1va35kI/t4YWf596MJd86ZjnFt91izW1PkeCNJ3XniXbOCa6PQQ9yGtk/qbEoCbnsTdlSekUjqPlM2qtm36+G2hgzzV3hCeQpeqiORHtWuNep9vV1sdS8nVUlyC54aMA1hK92eil2iG0IWKJdqtEbG/5CDkOWvlE76JbAMF7ERgG6XaAxcQ4DNeWHN2MrpTkWFsQB1NJSfreczZJjtNApp71HB0mAMp9Az313uNpLbb4ruc8o7WPqdPtU1gY2R0od0HLqX3DfqI2n4Q2Ri0TaPns2Fl7Pf38B9fAmHHHgw1FTk5trlNLl5pKbvFW3NBzjXsiV1Ge1m0p9b3n9K5T3PVQ2tpRoLTVd4GlgO6kzqRfRe/1cAs19qSEFOkkkUrr1yVr03tkrFuyVpuYdOTUBDx6V49kqKfesWwdonulJc/X8CNsdYTRn5r6BcCnNJByjlPARmxDRmaExvheBF6Gg53SGsA597veo34N10oiJKW9sYE79fs3BdwVrfue4USU9ZQkZT9zbLA9CF1yytrPaI+67vd0gQbGjDsjKP2jbWhOylcKSV1OfRviU4S1VOZuAqKE5Ka2yQ8nh7lXdVJR/VQ6FRBL3Vzqt97wrY9nTXDK+02Ka/HS1cU5O7ALa4masA8eEU/W/Na61lbnzQ4HKUyI+aHUG9rYRioy0v7IKDPgAHFEBka2Q6aDSO3T31DUhT1DrWyGeyjdTHEYAE6halbDlw1ojcDfSudj16FHa9dNImslS5QlrBb3Nfe7bnHt1uewb3IggXfH3it7LYdS4DQ1utpxHFrlwbr/HcUsbgeixJfBl/9f+xm11lM/30YamErLe7FeUEAh6tbXSkYymYK79vjh7+duAtb3eQWIMoyq/B0D7cPhZdu8hnXV94uIj48I/87UWN6bTZ9H7WN+Vanp10h+6P5XlmskgGapHsaO59G7fwZGZHWEISXn9GBqGLIZ2AZ65oCl4N1m9KW/1SUNjqlFG93LC4+9uvoQ7/Cuha3OMWrIU46XTVOq32/qzXR472rIHAzPG//BPYpyg+wX8GZLmftdt7yHludKHMzYeykQsN9wjjisQyySP4MFUf6zV9VOokyV79s7T3QscLhqWesZtdRPP9eGGpgy7onoFA0yUNcKVC2iBjVym8K3XHgCaXSpqKEs2QRwsDPwojGtJ84lGpE6xxjlIxNx1QbE9aW8sUP5XVKYDu9Lj2rSR+prBbzygcoWAPJxjFCkcpTWFPV/97oG49qa933q3Grk1mTsoWeh15cjo0TCiRHhyKYM69uQ856r7FGdUs5rMf7e6SwdOZ/3UyrUYKAURl9PMaaxKVrh5LZ3rdZTAG+l9+F1sysBcdIzq7/0Pd3yu96Lfur7kEJXJpJJY4DtOwwz5140LghQ8kk2zDxW9sd7zoDLSo21aZr1nbTYMlGMO+CpbqCpZ7RHHfV72kgDU8bdpoaQ41TP96lbZCi91Iy7zfRSj5/3OUYIs8S4u09pd9kEm7AeVNEghDtEq7T7MF3PeIjIcI6firqtT+9vPRY2Jz3lZLDWM8yn9M6BEMUC3GhDvAexaeEHgC6GXZBOl1ZnwD27YQpcaYRzODbcp9aHjZdzUEtOA8w/y/oow87AD0WkDp/CKRiOuL0H3ecaOOkcl+GQkz1813vUsz1V3Vy2jwMkC2V/YJjxhgxHCuvgkK4X0ft2p8oezmtfxg+QkjwXU89oj/rp97SRBsaMO6MsHck7ZOCvFchliFBRU6sRn0uNe70GtbFsY1uzPppsdj7uLRjsRA2yB1r1ULLeE3re87IJAi8OHbFr389Lf68MIwMgJT9M7as9c6BETjbWpe1hl97TLX6XTGmyJ2rFLaXld93yvlqdS1DhXfEtrlkuW/MZtdJFP8/GGhgz7vqC1Tvnsp9laxkg3VQ6+9plHnET8EFLGeML2Eo4LkBg0n8Y27biu99qfVteR5pUHVQJp9YjJwpYT03eu07X9yxKI8o+w8xGizUf8btuse7W51jzGbW+136+jTQwZtyBOV5W0N9Q4NeKFjB1ahuC+tMj1ubwnRtLCokNAbuliHC15Igq1Uh7z/X12pdylh5lfCDuU2S1fBvKTMotLZHj19/lNkfAIeAL0P1RZ7oe8bveRuPXX2XqGV1/pn7E3WhgzLirQeuxHPZT1ouG2hSxaPUaCqSynvCkDX3E2hzdie7oEoJ2a9HiJ4UqnVy36G19H0e9Hh4Dtf3h2FYsdKJ1Nfua1+Co6zx332rFSiFKEENOgEf8rs/p6xZ/f+oZ3eJ++jV3ooEx4y7axFNsgwOMG9Y4pZlFhVKV6BOHwxGgz0Wuak1LatpazBKopn8buOpppZ4PTbrndGgCu4CyZC+m6Gl38hr02xho4F0KmLL1TICjKVpZgmHXY63rg6Gf6ve/dG1H/q4vXeOWv1vjGW15//1aK2lgzLibmmWaGsHmpd2nFn3GUJu42m2CNbELdD1DhssbEc4SgVI2ZWtM5uIBltzPNce6P/3UBDDxja85uP/25hpIEp2tOh1uvuCJG9D58aLq76Tnl/byH/m73uNzWuMZ7XGd/Z6u1MBcbnnHQcN7seqJZFo50LNqX9Jz3aVr4Iga0HWgHVLnB56GLl0DXQNdA4fSwBzjLi0PNIdJSQ84Tu2cHJe93WPtQ4dSTL/Zh9EAYiPjXJ9XVgyvAAyJaKRVK+fDKLMvtGuga2AfGphj3NXB0bTm0BTkFghUtF6pL780Ip6zj+X1u+gaOKsBIDlOqjKS91opSDr+GQVUd/YE/QddA10DXQN708Ac455rUHtHfYguEXgOUxsmL/8sBd3sTU/9fu5XA0hoGHj8DlrbTKTz/0Pu7/vVQF9Z10DXwN1pYIlxvztl9AV1DXQNdA10DXQN3IMGunG/h6fY19A10DXQNdA10DVQaaAb9/46dA10DXQNdA10DdyZBv4/n8Ke5o64URQAAAAASUVORK5CYII=" width="251.5" height="47" /></span></div><div class = "S3"><span class = "S2">The aim of this chapter is to introduce a classification which would describe the properties of the equation and its solution. </span></div><h2 class = "S5"><span class = "S2">Basic classification</span></h2><ul class = "S6"><li class = "S7"><span class = "S0">Order</span></li><li class = "S7"><span class = "S0">Number of independent variables</span></li><li class = "S7"><span class = "S0">Linearity</span></li><li class = "S7"><span class = "S0">Constant/variable coefficients</span></li><li class = "S7"><span class = "S0">Homogenity</span></li></ul><div class = "S3"><span class = "S2">With this criteria we can describe our example as: </span><span class = "S8">Linear, non-homogeneous </span><span class = "S2">( </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="40.5" height="19" /></span><span class = "S2"> ) </span><span class = "S8">, second-order PDE with </span><span style="font-family:'DejaVu Serif', serif; font-style: italic; font-weight: normal">d</span><span class = "S2"> </span><span class = "S8">independent variables and constant coefficients </span><span class = "S2">( </span><span style="vertical-align:-7px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPMAAAAqCAYAAABmx23YAAAOGklEQVR4Xu3dc5R1SZYF8F2taducadvm9PS0bdu2V3PanLZt27Zt27axfrUi1oq6FU+ZL1H5xf7ne/XevXHjnoijfU5k7ZeBIYEhgT0hgf32xFuMlxgSGBLIUOaxCYYE9ogEhjLvkYUcrzEkMJR57IEhgT0igaHMe2Qhx2sMCQxlHntgSGCPSGBfUOaDJ/ldksNM1uzSSV6/R9ZxvMaQwD7BZh8tyb3KWl80ySnL52Ml+enYA0MCOyCBzyc5TZLvJPnPdT1/X/DMraw+nOQcSb6d5L/WJcQxzpDAChI4dIkUD5HkNUkut8K9cy/dl5SZ8ITbhPnSJFddlxDHOEMCK0jgrEk+Vq6/X5L7rnDvUOYigTMk+XT5fKckj1yXEMc4QwIrSOBGSZ5Wrr98klevcO9Q5iKBGyZ5evl8viTvX5cQxzhDAitI4PFJblmul+pJ+daCVcPs8ya5UhL/nijJUZL8I8nPk7wuyc3WMqutGeSJSW6e5O9JjpTksEWochYkxKGSfD3JC5I8OsnftmYac0fFvF8lCYsttz9mkn8l+UmSzyZ5S5KXJfnZDsxtM49kPK+V5AJJjpvkP5L8KskXk7y7pD1fnvMAKVKVy9mLXP6a5LtFJo9K8sPO/QjPN0++d9/hkhwyCS953SQnT/Yng7+Q5P+TvHjByx41yU2TXDLJqZMcIckfknwjCV5GLvz2JP8s45wsyVeXEKC5HX6je29ZZbYYj0lypiS/LxM1cfdTEGUfm4zAdys+muRsST6T5A5JXlQmSklsFpuEgsN7klwkCeFuF86Z5LlJLDzlfW2x2jbKhcrczeXPSa5WNsx2zW2jzzlOkmcXWf6plAIxuQdLcpYklyifGayHJrl750HnLnI5SZELRaHEqhFXTnLssievkORtk/tvneSxk+8oLCV8VdnPvXcj35fMeOlLlflwZB9J8sEkvyjz4BiOX+6jzBcun09bHJ33pi/wzSRvmjzDuv/fRoW9SJn9/pAkdy7W4gEl1/xjeaDJ/aZYEwvh2t0IVhj5xSN8KsnxktwtyXMa68naqjufq7yAcpb33Q4g455fjMqzktiELH2LxyW5Vfnif5O8YzsmtolnnCrJu4rSfbxEdEoxLa6Y5OXlC7KuJcR6zTWLMWBse3JRdvxAklMksScpzbc6c24ZZJ76hCUieEKJxkRA905y43KvMUWfUzAsIgnzMbfqEOp1nNp7kyC57K3rTQbgxRkTeHiSu2xCvge6dZ4yU1Qb7OpJfl2sGSvUQs32S+WLi5WQZ53zW9dYIopPlsGkBebKck5xuhLO+v5HSU5Q0oh1zaM3zn8Xj2KDIEOE2D0cOckvSzTE8AhTdyuOUchGIbWckPztoR6si9+nZJBohOKRCy/K8/ZwjZIa+e3Jjedrr5WyCH/B+t8+CePYQprFMVF8/Qc8/xTvK0rOkDKoPVynKPJtOs+Qajyv3ESvFoXzK63vPGVmqVDnXl7u0fMErBOFB9Ztt+ZyLYP4lAW5PSUWuoHQTji0VRDWs9RyduHziYsRmfW8H5TrzGtZ2Lg280aBrBFGrgKlPyEwUELKOAs4CgrJW36vXCSnlU/7TnhOLj+eMQDuppJIvLJrpxDa4kzgrWU/94b7fona7GP7uYWwmhzozLw9xCEgW5+U5CuTMeT21gM4wunvq8j4QNfOUmZhAkuGkMG+Cft6qJMjBF5st4LFRliAkGweGVG7c1zLa8qftwptTvfMsgnmPUte/5eS9y87J2GscHajsK7Wd1mcvpkfXgUHICeeBeST3LqV820LEeUeFYga/vbGELHUKAW5KaWa4qnNGPNSFKkYAkpagF9pITWrcnAdI9WL7ubJSdphT+GdkLCVIFtWtnOvm6XMmGmJvo0jocdW92ABzl+Y7MusZUZbM4giPQPFC8qr5gG5Ug2TvEn+tFVoDQfCbUrgbNVzt3LctvTyoCT33MDDeGU5NyziB1rPTEkQhlNQToQbpT/6DCUSjYnKAGl3/c44n0hy5uZ7KYJQWXr0tSXe0/MZn1k5+RJDzL6kp8w8lzzYby8siX5vBL/LMQiv18lyu/K9/PRDG5jlK4t1FI5MyaBVhpML/baQX7NyqjoeAsOGwBdA65Uw+uaE5FtHw4mxGQ7AmltkIeVBHbxxDXUXKWLvXady4cGkILPgGdUIirjs3xYt+fmG4qR6Y7VlrF6+6x7pjbRS5WGKzyURAWgI4QSncK/SJ8yLdje8/j1l1h2FaYN5FD2h1dogSl7JoAXanjXEACoLrQIGghWToyisV/Z8lTHqtSwpiwo3abpvemPxxEgOULcUWlUonWAf75hEerFZOLWl/AQ605BAB3VYN4azgoFi8FfBqnK5a1NFUU6yZ1u0nX/KPrigHlQ3Hlx+WNRUZE8juvACyMgWSp8MzDSadS0uAeTU0qq1oqfMrFxl6oQeal89yGNYImjJi3qt8BzBs0yxvDe+cBiruNmTTRQYYQH/U8ols4SoT/Y+5UelkBs0F8qllBZWNUyznqXBBkkC8tpKGK11gbd5sLa6YTNjtVdFK5dlehc00khRQGgsRG7hu6o488i4StrJY0UDIrRFEPVdvHBK2PeKnueVctQ6OsNdW4sXPWPp33vKjDFVUpDkH3HOSDWv3uiiLT3JTV5IkSk0IFtm5TZkgV1E2MAi67zJae3PalYP/4zSjbTZMXv3I5OmZM4qzxFyKoktg7YEKNw+6TI3Ta5ZRS4YZ/tV+cochejTKK6tz6saTGvd9fH2hfnaA/WY7CrTv39TJ+cQq4GpYyizCeU31eU1b0I9Za6MHjKAUvcg/hdiE+J04vWYofvm1eN64wpZ2jKIJgmF/c2gJS2EXDq+emjDIIV/rYcgRxbKgVyIh8aabha8xCvKIIs80LVL04S5S13mscPTeW0nm62Jo4aXvfJOOzeK4314QikbpYRWLosiltbbWaOHdRYF2SQspuzm1wOnpQ5OHzSCKJW1YBB5eGnYLI/tfDJCE3oMvLQNaz9NqfA0SnH4ks1wQ90/TiCRF+LaMJi/qVVGDr2xqdXJJeUbFepxFEFtkeeRYy4LpTD3y13Uhs9T2uXcT9CXLUV9RNQyrZbCIMbJv2BRpu19vpdOYCYJm2XnyTCqQHktNotelan3Pu5RlhGxLJMaeE8GU1ea+ipWtqek1qB2rWkP3O2dX7gSJTSgsDz0FPYQOWnnnOaxrVwoAG/bK+EwzJ5Ffg7NKPnoiWjhOXJ4yjLPsdivOrsAL1I5ozqWtktE7ry+gzY9mDZQmSNl7ZHK0jmdYmrhCNpZMIYKEydau8gOcG3PM9dmEReyUpL1yrLynPI8OXVN/HskWa0TamLXb7wqeEaKTImqtWobP0QDQpZFHqo9O1rb/oQ/lKPCxtOVIwTncXno6bG0SqJM8+g6hrCMVWaMKCilnlXOa2VRSTXf9Yg15J+5GE8uL5Tb7aCgWGNg0PXrt5GMNdXAoeFIA4cNOj3U0sqFo/DfLbDJxq5G2H7sdcS1OfwjSltyT34qLw7XQI+BZ5zl/5hq/QrTfcf52ZPm04smWmKw3UMMFcVkrBjzeWRhjbAYLPv6QDl3T5k9mKWzgcCm5K1ZOZ1EOqK8eD2TiapHkvGWtd1TDog82kiib05CHkrR5i5t94x5YalbpewtEsFXa8f66v8VKvHC5iyHQmqBsFB41WsEEOYySvI5p2qmUGNv2XwHOermmKd80hQdUPWAimczOsop5mWjUwQb2kGXgwooh/KdPSPC4YVFPEpWoisst019ixllp6lceFVVhroH6wEGe9CazCrpta2ePk97qas8ra01BmPy5owoPTBXHWZIMUDoyn/tH9/p+OIgzM29GoF6UWPtKeCcGDMGwTO1jZrbrIMddY5t/wPdIr8DYFbTCOtpA+kcYjEsBPdOGAglApweqiDgqgiK9IyBEHWZcLidlNAMGaEYr3+1gmKz+PIeQlymoaPt/BFJyHkdGqE83otFtDisPKKkLau0c2LVec5ZbLi8x9xYTCGd6OaBK2ievmShllBdWK2uSgaYWsao1qNXGHLHLz1jEuSZ8Ff6AtIJUZdNL0RehKlcKIKWzXeWMRa1QwqXlVpBE8qsY5Y1tWzng6WuxyftY7m87zgolRo93PXYo3fCos9jqO1fOb2Ik8PEEeBzpJS1dDpPHvYHR8IoOPV1IC++6NTUImH3fhdqIglsxurdVxmnElGzCA1Gg6VnMec1E6zyzEXX1nIdQzKP2aXIogpzrDnYorHH70MCa5HAVihzPeYlfHRKZFVUhrLX3khZeH91PONvF4RU8rp6VrX3XLJEriHbkCGL8vntmvt4zj4iga1QZoSYcLzHCi4j1socIhxaEsmZX7mCfLx20iwz3mavqT27GHzhTQ9INO8rP6wnzTb73HH/kMBKEtgKZZYDyLexzdjKKTwTkab/u8feOerGC05PYcm7/LbdHq/27Hqve8yQruYFbOpO/KmhlRZ8XLx3JbAVysyDIQok/I6MTQvhlWGUyCv5tDVECqu2KCfeLaeweFylEaUx5YFVe4337u4Zb7arJLAVylzbQb0o1rK2UtYXx9hWVppHazu+ao1SPbX2SO+0wOrhefNA6qlHDwwJ7DoJbIUyL/OSSktyY212LWrDyvj/QC0jxXHNkEAjgZ1QZgqsz1UThOK5P25QoT6tZQ7p1DsTOhZvSGBIYIYEdkKZFeDl0U4lqcVesHRkOVShLKXNzx9FGBgSGBJYQQI7ocx1eggyLZqYba1wDnXrmtoLfzpnhSUYlw4JrEcCO6nM63mDMcqQwJDA/hIYyjw2wpDAHpHAUOY9spDjNYYE/g3Wl91J95OOzwAAAABJRU5ErkJggg==" width="121.5" height="21" /></span><span class = "S2"> ).</span></div><div class = "S3"><span class = "S2">In addition, second order PDEs and some systems of PDEs can be divided into three types: elliptic, parabolic and hyperbolic. The type of equation determines certain properties of the solution and it imposes restrictions on boundary conditions and discretization methods which can be used to solve it numerically. The next section illustrates each type on examples. It is followed by some methods to determine the type of an equation. </span></div><div class = "S3"><span class = "S2"></span></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><h1 class = "S1"><span class = "S2">1.2 Properties of different types of PDEs</span></h1><h2 class = "S5"><span class = "S2">Elliptic</span></h2><ul class = "S6"><li class = "S7"><span class = "S0">Boundary conditions must be prescribed all along the boundary</span></li><li class = "S7"><span class = "S0">Local change of the boundary data influences the solution everywhere</span></li><li class = "S7"><span class = "S0">No propagation of waves </span></li><li class = "S7"><span class = "S0">No real characteristics</span></li></ul><h2 class = "S5"><span class = "S2">Parabolic</span></h2><ul class = "S6"><li class = "S7"><span class = "S0">Parabolic equations arise by adding time dependence </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="20" height="38" /></span><span class = "S0"> to equation which was originally elliptic.</span></li><li class = "S7"><span class = "S0">Waves can propagate in one specific direction only, corresponding to time-like variable</span></li><li class = "S7"><span class = "S0">Equivalent system of </span><span style="font-family:'DejaVu Serif', serif; font-style: italic; font-weight: normal">n</span><span class = "S0"> first-order equations has 1 to </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="38.5" height="19" /></span><span class = "S0"> real characteristics</span></li><li class = "S7"><span class = "S0">Boundary conditions are required in space, initial condition in time</span></li><li class = "S7"><span class = "S0">Local change in boundary conditions affects immediately the solution at all positions in space, but only in future times</span></li></ul><h2 class = "S5"><span class = "S2">Hyperbolic</span></h2><ul class = "S6"><li class = "S7"><span class = "S0">Waves can reflect from boundaries</span></li><li class = "S7"><span class = "S0">Equivalent system of </span><span style="font-family:'DejaVu Serif', serif; font-style: italic; font-weight: normal">n</span><span class = "S0"> first-order equations has </span><span style="font-family:'DejaVu Serif', serif; font-style: italic; font-weight: normal">n</span><span class = "S0"> real characteristics.</span></li><li class = "S7"><span class = "S0">Boundary conditions are required in space, initial condition in time</span></li><li class = "S7"><span class = "S0">Information about local change of boundary condition travels through space with finite speed.</span></li></ul><h2 class = "S5"><span class = "S2">Types of boundary conditions</span></h2><ul class = "S6"><li class = "S7"><span class = "S0">Dirichlet:  </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="103" height="21" /></span><span class = "S0"> </span></li><li class = "S7"><span class = "S0">Newton:  </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="105" height="38" /></span><span class = "S0"> </span></li><li class = "S7"><span class = "S0">Robin: </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="145" height="38" /></span><span class = "S0"> </span></li><li class = "S7"><span class = "S0">periodic: </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="108.5" height="21" /></span></li></ul><div class = "S3"><span class = "S2"></span></div></div><div class = "S0"></div><div class = 'SectionBlock containment active'><h1 class = "S1"><span class = "S2">1.3 Methods to determine the type of PDE</span></h1><h2 class = "S5"><span class = "S2">Second order PDE</span></h2><div class = "S4"><span style="vertical-align:-18px"><img src="data:image/png;base64,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" width="251.5" height="47" /></span></div><div class = "S3"><span class = "S2">Second order PDEs describe a wide range of physical phenomena including fluid dynamics and heat transfer. It is convenient to classify them in terms of the coefficients multiplying the derivatives. Replacing </span><span style="vertical-align:-18px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACwAAABQCAYAAABiZJe7AAAGH0lEQVRoQ+2aZcgtVRSGn2sHdgfqtQPjh93tD8UObDEQW8TG7vhhe23FFhvFwG5QsBBb7EYM7OaRNZe5c2fOOd/MnPnmyHnh+/HN7Nnn3Wuvvdbaa60xDBjGDBhfhoT7vWNDCQ+KhCcDVgWWAeYEvgJeBJ4B/qlzEXWoxPTAK8ACQex34Jsg/jawD/BYXaTrICyXTYGNgcuBl4C/gFmBQ4HDgIOAC+sgXRfhTlwOB84EtgZur0q6CcLq90fApMDiwLdVSDdBWH6nAUcBxwKnDALh5YEXgI/jcP5dlnRTEp4d+DJILgW83nbCkwC/AerzXsAVbScsv8+AuYCTgOPbQlgJLgEsFgSfB/4Mct8DOpmLgf3aQHgb4FJgphQZD9m+wP0p4ueHIynFua5DdzJwTDB4DjgYeBVYEjgCmAbYJN6fCJxQii3UEsDrku8NAo+Gm/4pRUihqLfJgg6o4qarStjvDXyWji1fBPggR3oHAufF83WAx0dLwhJ168W1wK4FRK4Edo93M1dxz1UlLAnJiD2AqwoI6+X0djoMHUdpVCWc3uo1gKdzmGjKDOinjDjCeKI0qhJeD3g4fn250OcsmT0jTva5KvRaabY1WAlt7hfAFBHA35cho0A0cysBdwBbVSHrt1Ul7BxHA6cCl8R1KM1pN+Bq4AdgReCtNhCeHLgN2AjYDHgwgnUPpG74Z2BL4JGqZOuScDKPVmJ/YCpglvButwJ6tvfrIFsn4TSfqYEZI/4tHagXLbAOHa5LeD3NMyTck5gqDBpKuILwevp0KOGexFRh0MBLuNZcbgVBZj8df0vJSnjgCNcolP5MNfA63B+x1Djr/0bCLsQc2WrAvICJkTcjODcLOWrIk7DPvEF4SxDGtOZ2Dcp/BI6Lm8SoWJQilRgbhRSTIw9Fbtdr+g6RwfGyuWNUixqVdhkdVk1MNY0DzEs0ijKEJXgLsC2welQ7GyNdlvCGcQC9Ca/fGNsKeQlrFt8B0wGLAu80RbqshOVnYk/TZ/rf/IM4Ejg9Rd601LJhafLWtCDwbCRY3LWuJrMKYXNq5tZuAHYKNlqOVYANQvI+9tmNOWxnC7K/AiYS3bGuqEJYEtsDTwJrZX7JLNAD8ezd2ImkOOPjaaPCb6uCC/y0K9MY0I2w7+cHzEzq7Uzs6TzE3VEeKMr5mnrVBIp0bc5K0z2RINTKjKjI2ImwzRo3RWElEYB5Mgsq5wDWM9xKM/DqaRYG3Y4RVpMsJ6ij1wDbhXWxAWREKCJsocVWAb2bbnnvyANbGFQnfT8PMDfwBLB2wa9KWOLCngnH2z9h2vWuETHtoBJO6jbNAHwCrJtjtrS9umzhwuyFyIMqkWTlfwHMu9mhYmq2FPIkfEFkIZ3Q068VyEJ1sXokutXdPHweQmE2Uw/ZCY5ZOXor0uWz/77JErYJw4y67S+mSHUK6dOd/NDOUTXy/y26bG/SK+FYe4G0vSa486AjsgHka2ChyC1PMC5L2MneixHXAbsUTHx29PMYYqpCLjIPmj13yKLMHDHAQ+uuFME6iPNJeiJkCae3ulP3iN5J+6l+ainyoFMxDL0eOBd4GdClWyQ3fC3VSpMlbPZcO6tqWC9OqpdpQpon27uEGfeLcthqt7UeOhVVRrWSuBZG6L6tjSTQkagmLkgUCivv0NkysALgttsRlcUZUfC2/2FhwNOfhv1r7sCH4bq13cKxb0STh4dJKSfbLlErpBbSbR2zrUznMhHyCLvFSsY7nNX49FXIYMe+NO2z+nlzZkavUUk3oF7MQ5bGZeH1fKbzOSTz/k5g8/CudmL1RNhBRl2ebg+IrQWSdiES1Hn4/qzMbLYYGB/PF22NSjgLL7TGFi7YoEd1cm7V44848Eo63XMxwRydXLMSsmVLAsKYwgjNhaifaajzSmfN+EsK5nlCUrKejwQGPi5Ek+aBfConmBo/uFvw40DJGFWpb/ZV9guJV9RxFd4VeyHcL4LZeW21UUU6dl21ibDxhUGWJV7bFXLRJsLG2ppTdTlrKkekw02ohHZYx/F5kJZw7v2uLRJOe08FVHQPrKX9oIkdaJ1K9LzotqjEkHDPEuj3wKFK9FvC/wJ0Xx5gud46LgAAAABJRU5ErkJggg==" width="22" height="40" /></span><span class = "S2"> by </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="14" height="20" /></span><span class = "S2"> we can write the characteristic equation of the left hand side as</span></div><div class = "S4"><span style="vertical-align:-18px"><img src="data:image/png;base64,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" width="186.5" height="47" /></span></div><div class = "S3"><span class = "S2">The PDE is:</span></div><ol class = "S6"><li class = "S7"><span class = "S9">elliptic</span><span class = "S0"> if the matrix </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="22" height="20" /></span><span class = "S0"> is (positive or negative) definite, i.e. its eigenvalues are non-zero and all have the same sign.</span></li><li class = "S7"><span class = "S9">hyperbolic</span><span class = "S0"> if the matrix </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="22" height="20" /></span><span class = "S0"> has non-zero eigenvalues and all but one have the same sign.</span></li><li class = "S7"><span class = "S9">parabolic </span><span class = "S0">if the matrix </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="22" height="20" /></span><span class = "S0"> is semi-definite (i.e. exactly one eigenvalue is zero, while the others have the same sign) and </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="120" height="21" /></span></li></ol><h2 class = "S5"><span class = "S2"></span></h2><h2 class = "S5"><span class = "S2">Second order PDE with two independent variables</span></h2><div class = "S3"><span class = "S2">Note that in case of two independent variables we can write the second-order terms as</span></div><div class = "S4"><span style="vertical-align:-21px"><img src="data:image/png;base64,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" width="300" height="49" /></span></div><div class = "S3"><span class = "S2">The equation can then be classified as </span></div><ul class = "S6"><li class = "S7"><span class = "S9">elliptic </span><span class = "S0">for </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="100.5" height="20" /></span></li><li class = "S7"><span class = "S9">hyperbolic </span><span class = "S0">for </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="100.5" height="20" /></span></li><li class = "S7"><span class = "S9">parabolic </span><span class = "S0">for </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="100.5" height="20" /></span></li></ul><div class = "S3"><span class = "S8">Remark: </span><span class = "S2">In case of variable coefficients, the equation can change its type in different parts of the domain.</span></div><div class = "S3"><span class = "S8">Remark: </span><span class = "S2">In more than two independent variables, it is often useful to determine how the solution behaves on a certain two-dimensional sub-space. One can "freeze" </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="40.5" height="19" /></span><span class = "S2"> independent variables on some fixed values (i.e. remove derivatives with respect to the "frozen" variables) and determine the type of the resulting reduced equation with two independent variables. </span></div><h2 class = "S5"><span class = "S2"></span></h2><h2 class = "S5"><span class = "S2">Systems of first-order PDEs with two independent variables</span></h2><div class = "S3"><span class = "S2">The general form can be written in Einstein's summation notation, </span></div><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXEAAABQCAYAAADvAEWDAAAeWklEQVR4Xu3dBfwEW1UH8IPd2IGYKHZgd6CCnVhgYYuBgYGFitiNHajYgYGF2I2K3YWKgd2KrZ+v7x653Dczu7M7s//Z3Xs+n/d57/13dvbOuXfOPfec3/md20SXroGuga6BroGz1cBtznbkfeBdA10DXQNdA9GNeF8EXQNdA10DZ6yBbsTPePL60LsGuga6BroR72uga6BroGvgjDXQjfgZT14fetdA10DXwKUa8WeLiDtHxHNHxBNFxKMi4rsj4q/7lF+EBqzbF4qIV46I20fEP0fEb0bE90bEv13EE/aH6BrYUwOXaMTvHRGfVT3/X0bEk0fEk0TEl0bEh0bEP+2pn37Z9jRgzX5zRLx5Gdp/R8SfR8QzlHn96Ij4/Ij4n+0NvY+oa2B5DVyiEX/SiPjiiPjxiPimiPi7iP9D4bxGRHxBRPxXRLxmRPzF8ursdzyRBp4nIj45Ih4cEd9XvG/zfveI+Oxy6rpHmesTDan/TNfAzWjgEo34lCafKSJ+rhjwV4qIf78ZtfdfXVEDQiw/XDbs91vxd/qtuwY2oYFrM+KU/l7luP3hEfGJm5iFPoilNfANEfFWEfEqEfETS9+8369rYEsauEYj/rQR8Zjihd+uJMW2NCd9LMdr4C4lyfkDEfHax9+u36FrYLsauEYjbja+MyLeICLerSQ7tztDfWSHaOAJSi7kqSPijhHxO4fcpH+na+AcNHCtRhx6QVhF4tOxO+WjIuLjmolzNH+bkcm8W7mHj98pIr7yHCb9Ssb46wWG+N4lfOaxnzgi/jEiJEFrAVcEURySX42IF4mIP4gICdUuXQOb0sC1GvGPjIj7R8SfFJxxTso7RsTLlpf19csf7xsRnzQyaw+ICLF18pIR8Uubmt3rHsz3R8RrRcTXRMTbFVVIbN+v/PdbR8QzRsS/RASPHVSxlScrsMUnjIhvi4g3u26V9qffogau1Yi/e0R8UcESw4//ZzM5HxgRn17+9noR8bCRyfuuiGDsoVyeKiL+Y4uTfKVj+tqIeNuI+NGIePVGB9b930bEbSPipyPiFUZ0ZEP/mfIZ49+e0q5Utf2xt6SBSzfiPK2XLsfnny0JTfr/gIj4jDIRzxwRCoJq+arKe1P9+Wcjk/anEeHzX4iIl9rSxF7JWKzf5yqnIFWbP1UVcn17RLxxRAirCIfUcoeI+N3yB5v5e47oKzd7H79JRDz0SvTaH/OMNHCpRpzxVp3pxUtRwfcVEXGfiHj/iBD/Jk9fvLJ62n4tIl64VAI+68h8Mv4qBcmDIuJdzmjeL2GoLx4RX1fmKZ9HaORjIuIzI+IHI+JVI+KXI+Ilmgd+y4j4xvI3uZEvHFFI5k58/JwR8UeXoLj+DJelgUs04jwz2OBnLxV7H1ISjuKad42IDyuwQkdlhh23Sh0PVaIv+eV6XByvOzLl7pVhFkUlD7yspbHpp4Esekg5YdlI3yMixMCdilRq+tz8g5D+SKnWrR/oEyJCroO8YkQ8YuRpefZCLTh3OAZdugY2p4FLM+KgZV68lytxbvwa39Fo/TlKnJOHPfRyemndgyjtZvSHBAdLJjx5fMr8u6yvAYZZiEQ8+48L0VkLIYQNV45PGHsoolpsvjZhm7ekJg++FWvpHyLiKSOi483Xn9f+Cwdq4NKM+FsUciTqEE6BA2/FMzPeTzfipWVFp+9JjH39iG4d5UEPefMMCu99SD4oIpAy8ehzczhwuvrXyonnfYomoE6gT1oRakmk0MeWEEt9De9dOOy3IuIFR7T6AhXsUJJbGG5I3EeZv42kDt/1yeoaOIkGLs2I87qS3e75IuL3BrTIE390+TuyJPHxWr4kIt61/GEKP/wbxQD4Db81JtAPL1OO7Yl0OMnkXuCPCHFJMgtt/H4p5GmRRR777Qs5lv8GCwQPTBFm4cGTqRoAEMTcwN3vq0f0Cb2E5lgB2RtdoM77I21cA5dmxBlnRloCSiJqSMRLvXBEoY+Cn1oeWRAt/1qO0kP4YaX7f1PYEdGiSpSNCaMBfsjr63KcBp632pghiN5h5HafWjxnpyThlxpd9IZViG2KP0dyNDf4F40Iye4h8Q4JwUEoWTNdugZOqoFLM+IMqzDJVAwzk1peOMUfNbe4OKj/l9wcQjXk5NToBoVDin66rK+BOkwCXfTxIz/5k+XkI08hX1GLHEcSnzm1fevIPbJS0zqxCaMw7tI1sDkNXJoR/7HCXDdWXec47hjOWx/yoGvYoMQYIqUhgVrJz3j2jtOt2ARerPxxl7e+uYWx0QHVFZS8ZOGwVp4/In67/FHs/POaC2rYINpaBr8ViJX8u/oCifJWbADCdylT3vpG1dmHdQkauDQj/ikR8cERMfbiSS5+T5m4oTJ5nnk2ixiCpvmqCk2VminCJYp+WoE/V2zy5RHxERHhBNDleA3IK4CHCpmAj7YCMQQ5ZE7kKh7bXMCo36v8TXMQSclabPScAYacyJEo+mnFhsJDVxGqDSCUS/fWj5/ffoeZGrg0I65Fl9ilxJcXKxNY1AIqxrhLVirugEJphT6EZMS8VQBCKOBXSQFd0/rNCyvmzuA/y4TOlWk79jP8uXnMnKJ+eaMB4RGGE2GVgqy6DZu5FZtGcDWGLKqrddvEtnnVFcrcahpCGHwdoYbE7wi//XxEvHyfqa6Bm9DApRlxOhTmUI3nOAxyyBPDPodhkAEQapHQHOM5QYwlzk3+qiRBvag8d/F28XCbgU2BtwdqSD63MN3V8wijLpE2Vbp/E/N+7r8pru1kozrTfDHk5haahK597lQ2JEJmkEVOSsSJCxyRA2CTZrDFwbMGQCjsD0uIhoGvBaWDRPhU6f6567qPf+MauEQjTuVi3rC9oH26nztWe3F5Xkrkp5roSm7y1lDL+h5yK14fQiUvKw/fvVpRPJRl+PmZk4DjOcPSZVkN6NojsZkoJJW6qjYZd4Z5SnjwjP+rRcTTFPSKsMrnlPZ9NgMQw1qGTm+gqMItuFesjS5dAyfXwDFGnDG8ZxnxVisWPZ9wh0Ic4ZFTSsbXhVGS1vaUv7/rtxivbCK961qfq2pEFKZHKQ6athJ2n3uscY1N0gZqbKfumZrxdaGUc6oB4FQoTELVK/lurVoPf1/Ch04mkD0S9nVIco356/d8fA1wMHDga+zOMUELAiLLkWRLOBqP54QeasQdI4UU8vvvW8IJfUIep4FsEcYzlNjcmvBgMzEI52xOiTCUuHKKReQlFyOuycB6COEWXUGuiKW3CdStzbfxOKHK0eDNR8EsHKgYzQnSPEvS25Ay1OQ7eIhUP6uM7VTL684qDibsqhwTJ0MbqUiCgjLv32Bh4aFGPKF8+UhfVlU5rvuY53P35FYRQxdX3bLIIWTBkjzCtwwM1ktuQ4L+SRFDhsm/RkluFR1/wAu3LvJANl5Je5uPvIH3uBXG/Z1LOPIpyoeqYm1UvZhpvVmuUW/mBm9Tio5U+O8VJmb04/8/PMSIZzkyFEfu2I7Y4s9dHqeBjKuOlf9vSVd4P5I6QFUkLP2Q8BB4bZKARNFMdjba0vOcYiw4V+RGlOMry9+yOAlmYZRckY14Ki/kWfACZdJeeEViv8t6GshKcacjYZSWTgKsFlLuVnDmuUZcJaPYjO+Bz0nqEC4/zOwQj8V6j73tOyuzF37g+ex6YW7ySXhYYqHm1L+Nd0p+pfI8labrgnSNAsIo2Y3gLBuMbFEPxvdpZWBzQ2DJD9T58ted2ZrPae4c/X9Me98hZiPhu0fELxZK0PyukmgveJdbNjQGUUyrbQ22Nf1AaCSaY6zAKcdc4+j9baxqcmvPuMZ4srDszhHxQ2v8wAL3lOuwBp2gGGTe9JzkL4Oi0KluNr3AsPotGg2wp8nG6QSEmG1vmeOJS3rwLhlv8C4xQaiPjJshI0JK1OUW6BqDOEVhOldPipjEJDMBOff7Y9ffuxQw+VwhE3jlmAiZSWinaHuG2/saJTsHSfpC+RwrwpReZDkJreWOFXFUawYNAVE9PBdRBGaL7teaWMtBy+Kt+nmF7MY6ainky3aKUFK3ihEfq7gTfb+mh5j6yZ3V3nOMOANtx5CNFwMn2fnEf1/z0drO6ZSSnB2Or46xS0LPhLEY8aVjk4qgkg0QauHBEysKWdSbls+vqfu78nw0sxk6kgt6TEnqLgUfzRh0S517qE0xl4wcsS5VH29RoC7YFQnV5P+f6qj1OhHx8PIg53wSlAzP3q7mXp5JE5LWERayTm78wfnb14gzRgw2HpC6l2RNJuRI6Wh5bcLjcSKRI+DJwndCo5gMbcOWkrWMeE3UpRel/28FhhhXSXKIgKUxXpLb1yCaekBsCI3ZSFV1Wuv4VYYKvw7RydJGvJ7XKcrdQ8a6xnc0Gk/ncKqjlqRsVuOaDxQM5yxsMOMtBDtFujf6jPsYcdcw4Hgp7thUJdo5sxTZC52ohXNW6tyxK78HwEeuJSmoi7pYon+WJERaw4gn9wf4IFHRWienhcqgVRx5eUo4QmCGQUqHeNbn6u5crkedwJArxPDCgeb5/yU54pc04pBGdcu6c8hXcQ6tLTLVUUsy2efAAmgw5J7OWeoOUjYn0ORZso8RFxPjVSoM4Y3V0sZIlaTjmeiyvAbWMOJgS/tWGjLgjrFOYxk2Wv4pr/eOSxpxiUhcPgT/D76YLSOkjLNml5zqqCUH4/NHRcQdFlwu4v7HkJjRubaPc6WGck5tXqP33WXEeZm8DSXXYjhtZhsdp1BCenLipUskZeYq4hquX8OIC48k58cQ5SrvW4m20MH9KvyqYgRJ2y7LaWBJI26jlZQkU/Hl5UZ//J2ykQdbA/Y6dNIDcWZvoG0UpEkCLyVCoMfcD0zwEIoCoaOsnFZ7MPt0t8uIJ5XqVGa7jr1hldOYtsvyGljDiIvtZnIFNS+SpzGxSMV/bezkEuKRy8/S4Xdc0ohDzkjGkgdGhHLuLUuNdJNvAY0cEp7yI8oHU52dtvys7dicbiVrcTvJPc0OU04ZcdwaDIf4310ntFKjG6YQC0pGTY7dZi4Z1T4dxXENbBmTvQuDTcVizeLUQ2IjFWOvG1LU16HLTYO87yL2wmTXGv+u4YND90hqXZ+JXyaawP+Lm/OOULge4qWbO97Q1Pe3HhIYajJR61EXKEfmIRGK1GlIok6P2CGBHEpkxtQcZ2s514ixjtHy7rtO6uuOeY/dxxoRxqvtgPBIQlWnil0ABdLRgBbKXrntc+yzlg559jW+Ay4JNukkYv5bAWFl2+TakJbdSqaMONgcvoU5Ih5uMbbieKT234DFsRyZ5sg+HcUvwYgzxOnpztGPa8XjLIZ9xZHU0dQRVTJTdlzl7ZQwIlli3nZ3z2PhoRWM+f37TGwC527EPVubV9p3vly3L9Ec0qpsarEkDO/Y9zi/b63i+E87cI9CX+AZp06E9clxKnyxz1qao/e1rq0rNeUEtBNsJW0fB4ojdysZM+KKeXjgjmW7QP6KgO5W3VnGeKj4AeUlSsUE6s9RTO8ofsupaEmcuBxHzu1UU+h6nupjekt6ZhOA3tg3UdrOv83LmJwOrlGWDKfU8d1DGnnb2G3yjvatw3XMe2xezbGOWNkG0d+yrsJ/23yg4YYEu6Y6CclaHuqYnMtaYpQzh4ib3jvVSto+wIJBJ2vIiItPOVbzqFUV7cIC3745/u06Ul7jC7rEMy9txHnUWdgjEQYHPSXJpJYnhV2FQUs88zXdY0kjXsP1IMuymGsffTopq/QU1oNw4f2vLQ+rQrYKqZzaW9EXAPkT+6T5hzjykqJyWZjnUJF32GUr23sDC8gjEpXYDPVsGTLiXmY7wpxjmJ0xMeKq2lRvEi88XC3Fk7nJiN5R/HFTurQRR9qUJfb7HNOT8MmIhH0wrVm02ZjY3w8hQoOxT0Oxy8OavcDP6AtLGnGGEEWu8IU5ut0eoTKqslELS/KGMVneqeCwj32PjaeG3wkbCB+kJGwQXzlE1JDct2o2PtYk+5i1dBPoFDlEcW7P7SRbo//qpu70Mdp4pzXiJh0GmOF17NmXBB6ndFZr8u54aYTxNoGSK+KBc/kbekfx9Yx4nUOQUJFYGRNzyEtguAkPAnKJWHyy6vIhwjJzuV2sOTwZ1t1BFWtnZKinhrqkEfc7NVYcDS0HakoUdqGe9Y4Kh6o8znLvY99joRlhVtTFwgbteuP5C8VJ3DsBtEVyQrZCf+5BxNAV/bRybmvJOwNAMkT1Sw+e56EFEKKQkF2+lbRGPL2qucYWGiF5JYbiq8m5wQg8euZLt0RHcTh23oXKNQZDPI5RkgDaerIs1bWkJ27eHVlvW14YRngs2QxRBImC7IpYVE5I9YumXB8x2j5hmaHp5wBwBA6pWEscu/CfeVYEoq3YIYUXM5fmopcvbcQZXsgxxXrWuGQfY94iwxhYFak+Vz0IGeP9N5+tHPMeuxfkDQNuvdXjEBfOpF3LHSMByksWn8+eBQz+GN3BMWtp0QndcTORCydPMkXk5SRlU1U5PSheZkbyAYVP2u7N+84CEN7WoPUvd3PEMRjJBv36CKSD8m0i3uPora0Qby530jnKOrajuAVjp0vUjCOLFzy9P9nwrVKJ1npawojrzOO4KnF1r3JzCJW2iztjaF7pHvyLmEfHWGul9ZQyvi48gz9mriTn9dyKtZqB0W9Kmns2z2jjcQI07nOQpY24Z2bInZoUkzjV2qgZUkUp3lMerk1aktB7IYyKZ2WMlfGY95itcV/EYeCFtYhvK0pyjXFAYmiA4J0VpwYttF6FGIg1Zg3qOdl2ljp0LZ16jeiK5eRJrGOhoFYSvWKTS+K5W11EaeJeQwH1XeDzlnejvbkJY7Sz6QC0yyE47iU6itvlYXRVJcpwWwDgeEI8CHUoMcuUTz2Z+/7esUbcC7tvRZkXifHzIjnqSiSJ3415t4kuOJRbW3ccR+S5FWvWoA0IZzb8cjZ+FgoARzPPEu01EmJffZ/6ujWMeD6DE7CNlhHk0Vn7nDV6gSlnCLXou1XXmEoJx77Hyeei49UQVh7u22bjVMeJEGpw6uMQGhdkm+RmLUPc24eupVPPd03khbp6qFWeE5INjTNtMx6UXRWbSzyY45OX7NDKsbU7ils4jpJgkg9Z4oGv8B48Cp4F731uhp66bBQMzUEVayP65l1ixWOohNLmNEO4winc+cjHvsd6uNooli4+age+xlraqZwDLkgiL2Eizu5QxCPb6o31vf2/nz2FEXcMYohV9yVL2ZxnXrujuF1fnF5skCc4BG+aM95rvBaJP88O3HSuJB8GfPhQxdrc+9XXC5WhSz4HKtZjnvMU3z32PRbKE369SxVGWHrca66lpccKPcRxmepPnA3MQb1VbN6YJ65MVkxrn7LudpCn6iieC2wuBHLpiT3H+8ktiHNKJgpZzZXkwxirWJt7v/p6GX1j44WD2c2lezjmty/tu8e8x3Qhfi2c44SUCb2ldbTmWlpyrDX761QeSQhVGNQJdRSAcQpPXPWV5ISY2mMHNGH39FA87jZhdqqO4qlUmXnJlNkkNEvO8JndC6+OYg3wMR7vkOC+4aUP5V7wvYhfj1WsHasO9AA2l0NPgsf+/qV8f9d7zJYwolAjQxzfYtpOa5J1a8naa2mJcUu6W5OSuTjf5QCG7CLIJ9CBE2pSKNyIJ56eNG+IITfYtnRUYsoL5tgLHVPLqTqKMzJCAqTuG1lXkvmMsUlUx5BCJVAR/IBcziZ3X2KF3MA9MqfACIOCDb3AMOCOhPpIOiLWks14D65Y2/HM2X1K4nOMC6imOM3b6dqTjHntT6AL1SAlBTTvkjnW93mPs9mvRCnDVDtCYKCSk6P8Hwut27XX0jHDZGMUUCruAZFUTMUBqpt31PcXuWDAwQ956wz6YJOZtT3xthnoEEjf7gnd4KHwMtRyqo7iFqnNRXy89thAJGXycTOrWnMs98KKZ7Vit4TiYMgUO50L/vyYhem7mvt6gYkFWRu3vLfSbwbUht2WceNaAVFtK9aOHVd+H3fI/QvSZixmL9TiFKHALRFUY0Re1gqIm7UtPGDTkpi65Pne5z3mpGHYlNxmsGokk1Z+PptEWSww4WuvpWOGaK0gFRQi4egJH07BX+uuaX5XWGUQPbS2Ed/3oUHTgPzbF3zpjuJT46EgHsPQQhMq0AiBDAHzYakhcHDOgErtW+m6r34u4Tp6hdevEUAggmCBEAVZyLH0s2bjC0bWUbZuP9f+Fmhi1gwYZ03s1l6bib5r7S07NE+Kx8S+2+O/Vnb6DEzRxx4776dYS8eOcZXvb8GII+fRJFVizG61ZkdxnjaDy5tmtO3c+VLzqCQQhsI6CpWcFHzuSCPkkh04eHHihQpNGIFzKS5ZZUEN3FSRidAS/eLmgD3PkunEwS4BO3NiEpLxMttMhfCIoyieGMJDnGLRdNoyTkgl64P3MyaJjfdvmN9rF4b7ngXX7R2CfU55ZKGglgTfRXe8rx7pfK21tO8YNnHdFoy4YzTDt3ZHcThVMbO6alQik0fFe0hjrnJK8U8rPAljJOK64rs8DyB9SQiL+ByKSk698BhuRhGNrpOVuVYKjjNC6EkVIXKfQ5EjjDfoak2Yz+t2YlLMJQ6ZvCFjDHm1TpSbi+kSJd9DoTOfJS3vXIqKU+v/VL/nPTaH5hIvjyIriWxkV9BfY3wnh45vjbV06Fhu9HtbMOKpgDU7iouJio0SXrMXG8eLBAMvkBH2+4SxTnrIenIYbN54xj+VK4vZ8+zhm0dxnDc6w9v6cbpGkwqhwItSBaqM/1BsPpwt/hsesxOSJCu+EJuGpJEQGMMiXsuwO4ntQh4llI7mxgyPuLgxS4iKAUs6dblFAxBlTj6QKvQkhGaOs8R8KT0tvZaWGtfJ77MlI77Ww4OXZRsnuzfPqfb66EC8No38FC1rVlAZK6RNlnU7LnY5rQYYCBuyLL5TFFIu6IdabBZCZo7x+3Y+kpTmxZMxPu1EDjiF1WGD02qg/1rXwIkqNm9S0Qw0TwDqxYsuyz50PEbonqRdU00tHBl549kGDSRS/K/L6TWgFFk4hrT9PnM05p/xdnrap8ep78mXQBAQ4YAhat0MralAxPzZpWvgxjRw6Z444y1sQmqe81bh2Nuys81U3NQxXbxP+zqyRoeRG1sMZ/bD0CO8b4JcCcNeK3UPQ5u0MNou8U6oKLQObPyqPtt4PW5ryW2FYXOplXf9fv+8a2CWBi7diGeXIkoRi33QiHagGUDcdBhJ3uyhS8VL3QdZjZecwBWj9+xyWg0wnoy05LQk6ZDUoTQ4dQU/+0iWiLu2ZWaEpMJ3kcm7fe7Xr+kaWE0Dl27E6zDJWHsjsEGoEtC0qQ4oEA5i51n4k+2lelx0teU5eWNsicIkaFQxKA5JcuJAxuDs2Bf+mbhm92wbDWcFqAInnXC6dA3cqAYu3YhrVCHkQVQFZrupWunJV+5vwi/KhlvJxrPQDmhrwebgxEHQCCTEw290Jq/vx22ewloQLgrFWhH6kr/grYudg5juK5j2NCkgNbGXeymvdwqAStqFdNn39/p1XQMHa+DSjThPDfGOSj1Hay9kLZ4fwgEcSimsZFkrvsdQ8MBq/HiNYoCAcI8up9NAUjIIhUGLtFI3mh3bwMdGC04KQmh98PglskEUed4aEWiwoPlAl66BG9fApRtxCsaJAacqno1fuhahEX0hxbgZgqzCzGv8TVk1iKKXt+bH4OlJcGX7MsUmOpF0OY0G4LOV6zOwEox11yKVl4y7uRma931GmM17Xes+4u68c6REyLEumStlH/30azaigWsw4kIfjtNCHgytF5EBlvTkXasYhHJoe/WBI6LHZah5dUOdYXBrZLIM+RMj7qg91C9vI1N+UcMQ9lA9a56comD3hbgU/MiBOEFJaB7CZVMjlpzA0EEg0LKxD1HqXpRi+8OcjwauwYibDc8prq0EGJcHL06VJgMM8yt2WotefsIsPHTFHGPNot1XoQ/EQgpWP2XlXU6jAXFq1L/QRXg5wA1xWoMUQiMd6jHXLHISo9aNRtGqQrt0DWxGA9dixGuFK5WG/cUf3hNTm1mKRw/EWrb5KoE/lIelHkRdY+DvwisKf5YicDr6gfsNugbSQ+2a6BroGri1BpT1o8nVkUrITRiFIe/SNbApDVyjJ76pCeiDuVENIC5DntV2GzIoRV+KuCCcJLWHrrnRwfcf7xronnhfA9euAZDTO5Umyhk7x3SokYRGIArBJDV7ovraV8qGn7974huenD601TWgcQSGQyX0EtQafGhgq7pTMlPrQEiXLl0Dm9VAN+KbnZo+sJU1AG0CnQTVAmsuya2rD5ZL1aDgp0OkWisPq9++a2CeBroRn6evfnXXQNdA18CmNNCN+Kamow+ma6BroGtgnga6EZ+nr35110DXQNfApjTwvwf8Lqu0EynAAAAAAElFTkSuQmCC" width="184.5" height="40" /></span></div><div class = "S3"><span class = "S2">or in matrix-vector form, </span></div><div class = "S4"><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="156.5" height="21" /></span></div><div class = "S3"><span class = "S2">where </span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAkCAYAAACaJFpUAAABNElEQVRIS+3VPS8FQRTG8d8VoiAaGoVKQaHTaGjUXkqFRESh8Q20Kr1eolCJBIXCF1AoRSM0EhIvDZUoZJJd2UzmujZ7c4XsNJucmfP8d545c6ahw6PRYZ4a2HbHa0trS0s78OtFs4jDb357GA/Z/BY2m6zdwUZqLt7hDLbRjclEQhG4jlWMYiha+2NgnhcEHlsA8+ldrNTA3IFmVVpbWqyRv1k0g3j64bXYw3LVa9GHtwRwBHdR/AhzVYEh/xX9kdAELqPYFcbbATzDbCQ0j+NCLLS164QTpVtb0FjCfiR2igW8YyBr9NPoaccOg0YABnBxvOAWYzjBR6Jo7nGT2R+a/Ndo9R52YS17FcL59eIZFwj37yD7xs07B5xjqgwwcTzVQq12WE09kV0Da0tLO/D/i+YTd85HJWinI60AAAAASUVORK5CYII=" width="14" height="18" /></span><span class = "S2"> is the vector of unknowns. The characteristics can be computed from the matrices </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="14.5" height="18" /></span><span class = "S2"> and </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="15.5" height="18" /></span><span class = "S2"> by solving</span></div><div class = "S4"><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="161" height="21" /></span></div><div class = "S3"><span class = "S2">for either </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="23" height="36" /></span><span class = "S2"> or </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="23" height="36" /></span><span class = "S2">. We can conclude that the PDE is</span></div><ul class = "S6"><li class = "S7"><span class = "S9">elliptic </span><span class = "S0">for no real solutions</span></li><li class = "S7"><span class = "S9">hyperbolic </span><span class = "S0">for </span><span style="font-family:'DejaVu Serif', serif; font-style: italic; font-weight: normal">n</span><span class = "S0"> real solutions</span></li><li class = "S7"><span class = "S9">parabolic </span><span class = "S0">for 1 to </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="38.5" height="19" /></span><span class = "S0"> real and no complex solutions</span></li></ul><div class = "S3"><span class = "S2">where </span><span style="font-family:'DejaVu Serif', serif; font-style: italic; font-weight: normal">n</span><span class = "S2"> is the number of equations. Note that there are cases at which this method cannot determine the type of the system.</span></div><h2 class = "S5"><span class = "S2"></span></h2><h2 class = "S5"><span class = "S2">Systems of PDEs with three (or more) independent variables</span></h2><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="195.5" height="40" /></span></div><div class = "S3"><span class = "S2">The characteristics are always one dimension lower than the number of independent variables. That means that with three independent variables the characteristics are surfaces. The solution of the equation</span></div><div class = "S4"><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="198.5" height="22" /></span></div><div class = "S3"><span class = "S2">now provides the normal vectors </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="97" height="21" /></span><span class = "S2">. One can divide the above equation by </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="17" height="21" /></span><span class = "S2">, </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="18" height="21" /></span><span class = "S2"> or </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="16.5" height="21" /></span><span class = "S2">, provided it does not destroy any solutions. The classification is then the same as in the case of two independent variables.</span></div><h2 class = "S5"><span class = "S2"></span></h2><h2 class = "S5"><span class = "S2">Higher order PDEs</span></h2><div class = "S3"><span class = "S2">It is always possible to transform a higer order equation to a system of first-order equations by introducing auxiliary dependent variables. The methods above can then be used to investigate the type of the equation. </span></div><div class = "S3"><span class = "S8">Attention: </span><span class = "S2">Transformation to the system of first-order equation can create </span><span class = "S8">spurious solutions -</span><span class = "S2"> the solution of the original higher-order equation solves the first-order system, but not all solutions of the first-order system solve the original higher-order equation. Therefore, the type of the system can differ from the original equation. Second order PDEs should therefore be prefferentially analysed in terms of the coefficient matrix.</span></div><div class = "S3"><span class = "S2"></span></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><h1 class = "S1"><span class = "S2">1.4 Method of characteristics</span></h1><div class = "S3"><span class = "S2">Characteristics of an equation with </span><span style="font-family:'DejaVu Serif', serif; font-style: italic; font-weight: normal">d</span><span class = "S2"> independent variables are </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="40.5" height="19" /></span><span class = "S2"> dimensional objects (curves for </span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFUAAAAmCAYAAAC1Q9c1AAAELElEQVRoQ+2ZaQhNWxTHfx4eCvEBPVP4QPLJVE+UeZ6HfCDkgxJ6JL16pshY0kumEjIVes88P16vzGVIUqZSZuGROUPoz7p1Hefcu++553Tv1Vnf7rl7r73276y91trrlCGRyAmUiVxjopAEagxOkEBNoMZAIAaViacmUGMgEIPKfD11KLDVY9d9oHYMtsatsgrQD+gFtLQ9VASeApeBI8Bq4EE2Q/KF2s2MKA+Ms8X2mHHZ1i6W/8sCC4CxgMC+A/YBF4H3QBNgIFAZeAVMMriB9ucLNaW4FXDGfswCZhcLMQc7BOuFjbtqTnLDM+8XA93cno/JBDYqqOOBZbZgHzPAYT9FMSQdqqBdCLCqAXAN0Kl8DjQEnviNjQrqBmCELaC3mjXuFAXOr0akoJ4DdOIyicKCYq5kFKB9fydRQdWxaQzcBeoWETAXU34G5gOCujnLhEXAFBujODw1LNRawERAx7oR8AFQzNFbWgpUBf6HL1fe3UB/l52U6JjFwGSzXYB/DwNVR3qFHZFPlhHvAfWBZsAWYB1w0JTPBOaUKDAXs/8GBtvAYUGenen4q3T40xScBkYDV9JW7m6eeRLoYM97A/tdrCvBMSq9bln9+gyoYyWWc0xV/XkA+MmyYVvgtQ+IJcBvac8VKh6WIDAXk5WglKgkfwALgyb5eWoFu0GoZNCRb5GhzFD83GnK7wD1XKzLMkYnQKcirCwHjoWdHDBPzqXT2ho4D/xqFwPf4X5QtaG1NlqxsmcGA7sAh+3/XcCACDYzwRJgWFXKA5vCTg6YNw2Ya3WpwHovB99M84Oqt9zORo0ENmYwULWaEpVkhi0c8X4Krk79gB3AW0Bh8Xg2i7xQU+VROZuomlO1Z5DoOqqML1HMURz+kaSjxVFxUqj7x2VzXqhtAGVziUu3Sc0T1a+SmsAjl0VLZIyA7rX6W96qLpWTeKEOAf6ymUeB9hm0KHir6K8G3Lba1WnREhjUFVCO+Aj0Bf7LxWYvVMXQ9aZA5UPKC/10dk57e4o5g3JZOMNYxS3ZEVZWAifCTgZ6WAzVzVEhza+SUItQ7NRY+U68UDsB/9oolRAKB0GiIj9VGUwH5uWxkfSphcz+urxss56q9hb0ci7ZCfXtc3ihVrfEVMnqMHWcdMS9orIlvUOjt3soIqiFUqO4qdD3xrxVThUkOUGVklWAmrASHevhtpB+q5eoG5SaCvrMoPu/pAbwuFA0IlhXoUt9DO1vDXA2i06Vj7oYOXmqdCnxqAJoaooFT4soMenbjTxXTYXt1rW6CaiBW6qizyXyvFQZ6bqPwDZnUENFjVv1ClUNqCOlbzPXAXVplAiUFV8aaMFNdW5cDSqmcbpyngphUM5QQ6yRTEkRiKrznxBNI5BAjcEdEqgJ1BgIxKAy8dQEagwEYlCZeGoCNQYCMaj8DLTDsCddxpAZAAAAAElFTkSuQmCC" width="42.5" height="19" /></span><span class = "S2">, surfaces for </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="42.5" height="19" /></span><span class = "S2">), such that the propagation of the solution along these objects can be described by ODE (i.e. partial derivatives can be replaced by total differentials). The number of real characteristics determines the type of an equation. </span></div><h2 class = "S5"><span class = "S2">First order equations</span><span class = "S2"> </span></h2><div class = "S3"><span class = "S2">Consider a first-order PDE with two independent variables</span></div><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="106" height="38" /></span></div><div class = "S3"><span class = "S2">which we want to replace by an ODE</span></div><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="50" height="36" /></span></div><div class = "S3"><span class = "S2">where the total derivative is defined as</span></div><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="112" height="38" /></span></div><div class = "S3"><span class = "S2">on the characteristic curve </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="58" height="20" /></span><span class = "S2">. Dividing the original PDE by </span><span style="font-family:'DejaVu Serif', serif; font-style: italic; font-weight: normal">a</span><span class = "S2"> we obtain the same form as the definition of the total derivative</span></div><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="94" height="38" /></span></div><div class = "S3"><span class = "S2">thus obtaining </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="91.5" height="20" /></span><span class = "S2"> and </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="90.5" height="20" /></span><span class = "S2">. After integration we obtain the characteristic</span></div><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="100" height="36" /></span></div><div class = "S3"><span class = "S2">along which the solution evolves as  </span></div><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="83" height="36" /></span></div><div class = "S3"><span class = "S2">The extension of this example to the case of multiple independent variables should be evident (see Appendix A). Finding the characteristics of higher order equations or systems of equations follows similar arguments.</span></div><div class = "S3"><span class = "S2"></span></div><h2 class = "S5"><span class = "S2">Second order equations</span></h2><div class = "S3"><span class = "S2">Let us now consider the case of second-order PDE with two independent variables. </span></div><div class = "S4"><span style="vertical-align:-18px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAowAAABQCAYAAACTUCSrAAAgAElEQVR4Xu3dCdg1y1EX8MINFTVgFDWyREDFFRNRVERxASSgRhMIBgSDIYlBRFASISwqymYgQKKGsIbFAAYVhZAQNYIgARUh4IYBTBRQDC64BBSV58ftul/f+WbmzDkz57xnqXqe+9x73zPT01M93V1d9a9/vUGUlAZKA6WB0kBpoDRQGigNlAZmNPAGpZ3SQGmgNFAaKA2UBkoDpYHSwJwGymCs76M0UBooDZQGSgOlgdJAaWBWA2Uw1gdSGigNlAZKA6WB0kBpoDRwUwbjG0XE+0fE4yLi10XEwyLidRHxjRHxlyLim+t7uEgNvFlEfFBEPCYi3jYiflpE/PuIeGlEfGL774t8ser0fRpwiP19EfG+EfGbI+IXRsT/jIh/FhEviIgXlc5KA6WB0kBp4PQauDYP4z+IiN8eEZ8WEc+NiP8UEW8fEX81In55RLxfbTin/8hWPpFx+N8j4v9ExIdFxFdGxI814/F57b/fKSL+5crn1O3noYE/GxEfFxEvi4g/ExHfFRGPbAcDhqS5/PTz6Gr1ojRQGigN3I4Grs1gfGVE/MeI+P2DIeSV+s6I+B8R8YiI+F+3M8QX/6Y/PSJeHxEfHhHPGbzNkyPisyPi5RHxrhf/pvUCNPBJLUrwS5tnMbXi4OBQ8FYR8Y4R8Y9KXaWB0kBpoDRwOg1cm8H4+RHxdyLib4yo8F81LyMP5NefTsX1pJUaYCgwDsAMXjNoS7jyB5qXkWH5f1c+q26/ew388Yj4RRHxrJGuPD8into8kH/+7rtaPSgNlAZKA7ejgXM1GH9KRPyWiPi1DcP0gxHxrQ2L+P8PHJ5/3MLT7xwRX3dgG3XbOg38rIj4nRHBe/TGDXv4ihZ2PKRlmFVeY+Hqn9kMx0PaqXu214C15Vc0byAMKhyiQ5tQ848e+Dg45D8dEcLWf+7ANuq23RrYep7ufmJdMaWBY+yFpe3SwEEaOEeD8edExLc33JKX+t8R8UPNcIRn+mMRwcjYR35ywzP+1BaShokrOa0GfkNE/MOIeMP2WGPAcPh5EfG1EfG0iPjePbv0qHaQ4DHmOS45Dw1YV14cEX+wdef/NajIw5uB/7ER8VciYt/DH/wqHKNDx75rwHlo5vx7cYx5ev5vfZ49PMZeeJ5vWr26CA2co8FIcTaF92j4NNmRQo0MC96Fj4iID40ICQ9LRVtf1fBRH7n0prpucw0wFHxzL2zhZQaDZKRnN4Pv3SLim/Z4anqcfk/zXO1xa116ZA38koj45Ij4woYxdThwWHhiRHxGRLykZUIvhRH83Ij4/pYtLXu65Hga2HqeHq+n19/y1nvh9Wus3vBoGjhXg3HuhZ/RNqLHR8RXLNAMDByjk3EiY/pHFtxTl5xWAz8pIv52C18ao+9e8Hhh7VdFxJdEhOSXksvRgKQVjAYynv/Ewm5/TkT8oYh4dET864X31GXbauCQebptD6q1XgP77oWlvdLAKg1cosEI0/HaiBBmlv38X3ZowEaDvw8m8t+u0lbdfEwN/MqI+OctPM1jOBeu/Nktcem/RQSv5KGYuGO+T7U9r4Evi4j3jojf2rDJc1f/4Yj4vIh4bER8dSn2TjWwzzy9047ewMP33QtvQCX1isfUwCUajPTxCREhtPwxEfEXZhT08S2r8ndFxHccU5HV9iYakA0t3IhX8RsmWpQNLRMevgeVDqOx5PI0YOwkwPy9iPjdM913ePibzYvMm1xy9xpYMk/vvpe30YOle+FtaKPe8qgauFSDUdhS1vO/a8kxQPVDQf6LogNAvozFo35GmzUOowqXyDBAsj4UGLi/FRHwbGUsbqb2O2lIePO/RgRv8S+LiH8z0gveYxRZT2nfxJ10tB56nwZ2zdNS2ek0sGQvPF1v6klXrYFLNRjftGVdGpxfFRH/YjBKQNuwUYxFOLcU3grZuUoFpvBmDXkZkX/j+BsTyTcqyJAviIgnXfUXctqXE6IUqqTfXzAISzMWeZp+fkS8SzM2snfKBvJWgSqk8D4Pufq0/T4TrwQT+9fbb3+kJeac9u1v72nmLeqdD25Z070GHAiMN1YEiTMp5vuviYgv7f6G/cC8zgz8/EnbqHzGBJG/tsBUJOiULNfA1DytObdch1tdObUX1r62lYarnQc1cKkGI+8E3BoMB2MBTrE3FP5kRAhDf9tgrBl4ahB/dPd32EaZm5JjtEUYH4zLMWGsoIEhniPjs2QbDcCzod4hklpe3f47jUVGpPDlELdq05f48ne7bnxARKAIYQzAsBIwBpVExuQvRsRHtR/UIUftVHJcDRgv83ToUTbHeJIZkuZsL4x5Y+1bSXGIEFEgT2iMCqo58V6ORR/AGvB3wkF7zh847mteXetT87Tm3OmHemovrH3t9GNx9U+8VIPRwKDYUBGCFyk3Cxu+jZ+hOJZJ+Q5tc+oNxhxk2Zf/tP0POhB1bMcErc+ntB+qasy2U0RoMsctufZ4j2zqjD6G+liS0+9t5SB7gzF7pqTgp7b/efeIeOlElyVTeAbeT8TFyMBLjquBv9Yyn3seTQYkCqz/3B0e+l44ABib3mDM361nvo+HRcQ3R8Rvmui+g8S3tN+sHVU1Zr9xHpunfQs15/bT59qrx/bC2tfWarXuv08D524w8iAKK+HqMyks8j/W3kKyg8QHBMA8EYT38BfvGGcG5ZjB+Ec7TyX6jj7k1TeZm5ws3jeppIuDZpWqLLx4DH71gf1Dn0nErVFhLyHit5nAtw0fzCs1ZjB+UYeH9Lz/MNHjXHRRMDk8lGyjAWvMW7bxVu0FzybvHkkibqFp4WHyxY2fce7pICVjBuNbd17pz2pk8GPtwET6nag7j9Kp5H4N7DNP+7trzm3/Ne27F2YPal/bfixutsVzNhjfqy3qjLIUSS5Pj4iv6QzHz2xE3msH8S+3trUzh31K3NX3RIQNqmS5Bix6PDpCw8KBKQwAmzjcqKxZokoIDNtaQdWDCmQOl9rjgNC3WGRL1mtAac8XNf1na0LFSvs9JyL+fsuIhzN+u/WPC2vGl7d2YB/Vnh4Th0y/k7doyXMbPP5qmlg7T2vObfsprNkLa1/bdixuurVzNRjR4aQXkEcCVtCmYuN/Zqsb/J5t5NSUtQGtlaSKmMM+/YwGrmfsyN583NqH3tD9MKLCwb+jvTPDDCUEQw5U4BNbOFGyA3Edcuc10o/XHC5VNm6GqiVLPXfNQ+ven9CA6kqI9eFPjfFTmweYl/d92++iAY9otd3VeF8rSTGiHfRMr5xo0JoiXK3kqCS2knsaWDtPa85t+zWt3QtrX9t2PG66tXM0GLOMn4HhgVAaSSgrRZ9hjtKg/JA9ywSODTjgsCxLIZg57BPDJjehXRyQN/1hjbw83Cf8J5F4Am/ak3PbaIQGk5NPJqxM1jXCKMhSg3O4VIeQTIaZ44Bc05dbupcRyBMPSwgmAo86pM0xzi9vSmFYylJfK4x+xr9EFwkvDn9DMdd/OCLeaAEH5Nr+XOL9a+dpzbntRn3tXlj72nZjUS21ur7npAjGoOxUxgKsokzZseosvECZnbyFJ0oIOql55rBPvCQZ5pJoAZxfslsDDIjvbZno39U8xWM1hHltZaxKapB4IgFljQg7Cj+SOVyqsCm6HQYsI8fhoeRwDfDQ4kAl+DTHCLeFqzMTfasoAU8meIHEKVWgxgQeOql2JEPhFCx5QANbzNOac9t8TVvshbWvbTMW1UrTwLl5GBmKyZuIew1Nw5h8bkR8YPsBifOu8oC7BlyIDNiezGGf1L59WrvuzZv3ZFfb9XsELzCsKTFunz+hFDWk36oZEpJi1spnd3Wm53Cpkm4YGJ4vyabkcA2Aa0gsEup1SJBRm4lqfavK/SW/okOCTPg1IrzNm0nm+DbR7mRCmz7kvF/z7Gu5d4t5WnNum69hi72w9rVtxqJaaRo4N4ORMcEYJBIP4NzGRJUXDPd9duWaQX12RPyp1gD+qgxjDtuUQcuQeV0jkF7zzFu6l2FgcyYShSQMDUUIUaiQ9Jnva/T0TyLi10fEj7QQ5Bgn3xs3Chdz4cURP5E4UXK4Bhj8DG8iW/b9J5pS0Yd3j1eXZ2sqe31pT2CalYwk4A4wsWMi2QYmmvzqVr986TOu/bot5mnNuW2+ki32wtrXthmLaqVpYEuD0cbwwtbuofi+PtQ8hSVDpfODDUyvjrRnrZXEPmlnymOJNBr1ClwI+hY0LucqdKTs2tLxhfVSXQUPJaLk3Hi3er8+1AyrOBaOztrCngnfltnSh/bBOKFv8by5LNw+sxYuFu3StYrvAc8hXZtfvHJIrxnSeA95Wn0DEoTU8h7zDO7STR9qnlsHEozvOfqyVvCmppE4l2GfFV4cIsAexr7FtX3Z6v6l8xiEwvhZn+gV9tuaNnZAmuvb2nl6jXMO9GgL2iVY/H3W1S32wmvb13y7S+fErjm4lbNp13Ou6velBsWulwYgh03jKSCHgthtZsmlN1VtQ5UHYQ+yRWKEdpIqB3ZOluCYoIKRhUl4R56xSyl3+DuqkOwfEDovG7GZ8JKmoM9gMPCq9qUQ53Cch7xWZvoxYnuapL6t5N/jaRLuP8RY6dvrqXIkV2T29bD/jKP8Dcj8JYe84AXcIzzF84ZpgG4ldwkZ4zPlZcWDiNom1wQwD2PC2ztVXm/stfsqKlOVkGCTrRcE1hH1x1rpqXLesX3rwzZlTpsDRJTiN6596JHvXzKPrVfC/zz31sMcP2PmAGQtXipr5+k1zjl7UlYCwoqR6yQ+3iVQKAdSerE3/sDSgWgHu7V74bXta9Q3NSe+bmGSpHEwnnOQtz2G6bYu3cpg5Ol7Vqe6Q7FgjAkGg0VwbPPWX+Fi2cpb0tokbxhvAxqQodeBJ+Y7OmPH5mvBuATBS5dhVgsevQ2F4cgYzixmv2/h5cvnqKCikgrh1emz3v2NVxf+jDdwK1obhjBPNLGYjNG29P1ynXHmpbkmYUyAeSTbgMOOhDHG+1DgDSV1JfWR3w/x4iPYV01l6mAlI11mOl3DjL5+A4X3fHNjiXCwlcpOMhqJQyfuz0uRJfOYYaLKyodGBMOdmNf92jz3vmvn6bXPuUyqmsvCH+qX59f3vaugxPC+LfbCa97X6KufE0r5OvzvkiQy32qf2fW8q/p9C4NRFQenWWFNGz9Zk22a5f1sXEmum0pXR1bCBKwb78BY+b9DBigrTrh3CMBXigy2zSIBN0l4aYTvLkHQmWQiB3wZr9KY2FAtiA9vPwrvZW3lte/pOzOZhfFlI0tKSPGbMZXgJGuWR1S4cK1o12LNe8ZAlR37fV2jDOJPb/QrTq2MS7CDaxKeWlyWxh3pPYNgF1URQwN1VBJpH5KQIsSs3J91wVzp6ZMkH/FyO5jNZa7vOw4fFhGf1m5iECdO0Z/gY1/QvBO86UQBAElslyL9PLYmjbFH5LsoWeqAJvJDGMYZlZl737Xz9JrnXJ9UNZeF3+vXOMFrC2mrKLSvrN0Lr3lfo8t+TsxV8er1ngfLuVyFfcfpZq7fwmBk5fNcwXqkF4kCf9tELdhdylU7mIGGT80kY2gwZoCAhZ0YpjBKazFufT8YMp5DH6hcYE0YF49snhJ0OjYYpxjC0OCFVHFmy37s0s2+v9sohRy9V4Ye59rgRZUIQCQH8FZsJYwyOmaQ8ADZ8PSPt4luLcKMuMx03eK5PemtRCU0SHCN4A5O8DyvQpM2Vt4u9DrkeTs25C36duw2GL8q6AhVOmDx+GUYeNezGfWpi0MrocAU8m4h1TcOjEaGpAxli7vfsyb7rv4s+Z13zSEuD628yg4gDkC+K8ahg0jybVpjXtN0wpg8Z+nnsTBovuNcn3sMHIyjQ4M5sEvWztNrnXN9UtVcFn6vX2upgwuYy1hkZ9dYrN0Lr3Vfo7d+TsxV8RrqWHKrQytGgC0iG7vG8Kp+X2swpifBgsuoSpc9Ja1x+eoX1zF8EwPDoo9UW21hnG1TXrI1g8PghfvjWRGitZk4GfJWMCaEyoceqKG3bM3zj3Evo93GSabCsvnc3jvgb1P4szX9FHJGui48CkfHIGfIyobn0RyGqtc8y71A+DxPPNO8rA4DvF7gBHCanj/mKYZT8i1fsjCOwTrIvvQxDDrfvEQohtihot6zkDYjhIhGwGUxJPO7PLTtsft4MxksvnvgeHOWhxWlk2QexipanV7GIhlb9mmLtvp5LKEFrm6XMDZg5jJiYH1eahivmafXOuc+tu099A7PngePXeOw9ve1e+E17mt02s+JuSpea/Vf93caWGMwWhhQKHC7wz7ZXPqsrK1q8lq8hBVt4Ptm/V3KYMOa8H5kcspW/YZl4g0l/s14mhLhdt62FEkQSWa+VX/6dnj4fH88Jn3I8hjPOvc2tx5/4X3Z7oShxLu4j47NbYvwtw1wrYfqUYSAEW6NWEvGfmgfTnHf1uOYfe7nMaqUHms8914SXkRjCJ7LTN7YRxeXNE99YyIUDghbRkfoS137xzbFvXtXSnSoy+QI3iohs2//EvfCU8yJuSpeDo6cEiKBWUWs16m/q0TVyxwXs+jTB7eLORw4li5V5II4QHP8eRdznY3FceOwLXzvfR+UNQZjgkdNTOFLwksk1ETglB59qZo8cb95vRiMW5BV911Hc5Q8eIyIJEoee71+QTx0czmx2q7mcVuOP2NP6FkomjyxCy9fjcLO9EW2HMepebxPwp1KNmk4zZU8PVN17t0tBiOv6hDDundDIzcwDNJTPpfxDP/7qAZzWcv0sEW/77qNU8yJuTUuS1VOQaxEQDikREMStz1Xhx7MB/5xKTTkrvU/9nyOKdW4vKewPKMZHtR/gzFwMBAHUwfUB+VQgxF+wAMoDe8aOhry3l1Cw482nEH+do6KO5c+HWtS4R900iUmQ1bR6d/bCUw2a2aM2lgkR0gYKTmNBrYcfx6lpFJxsraJbpFEdBpNXPZTthzHXhP9PJ6rWDTUHk7RTFzTN/desxzLYIQZ/aGmuF3JceacRBdGY8kDEKBjOEP6OTGXhJoE6GBJyRM9Ni7gZ8L3cxnw7CV4cEwfrxjxTF7CeINTMJJFnBiD4EFZMEP/+2jGfWwThxqMXMDwfgwLiR8pPb/anJFyCYo9ZR+PMalkoUrwgMckTt39iRcmFBCeOxqN0bdGxOc0CpZrDf2fckz3edaW4598lp6/JfXUPu9zq9duOY6pw34eCxU54C2dn/CZsIsEpRDc1zXLsQzGnh9Y0uUUQwbvPkOxOP7ufWXHnhOeJJo5NSfgsXmERTt7DuLhPMAi8WY76tCD3iUzy9ZJoaeYlxlOZ8BzLPS2Wz4/9w8GJc9rb0wurgTSv4xQF2wb8HqC6vP3zMblgSS7rPpTKOkSnnGMScWtjA9viTAWv7bR2yzNpF3Sbl2zTANbjb/5B3uSiSprEs+W9byu6jWw1Tj2bfbzGOk4UvKl0mPKJXoJZ1+zHMtgFJrbJ6NfJm5SPF2zvpe827HnxJI+cJTwCop6jknPHzqXAd9HUEG9lD7dRzC/PGmfGwbXwhQ6+B0icJlYZshcsq7xQkHHDvDvh8ghHkZYN4YiyoAx48ILwQOQXYkWh7z4Nd5zjEklxCwTmIyRFPMqOn0B/H5cy2A1sWBQYZ9KTqeBrcZfFnLPzzeHxTnd293Ok7Yax15j/TzmIUAHskSGhwd1u699Xh/LYGRs4wwlj5+pnoPBQzY1HFhWaVkyVtd8zV3OiXRgvbajixvTdV+Wdi4Dvs/RAMVDQ7ePYH2BHTxU9mW7yOdgCNFXjDNzVfj6soujB8x9DUbGBYCk05NT1JjIUMtFDaVFXzXiUEVd+33HmFR4557WFDeX9eUSBM/CLEn0i/gX8XLJaTSw1fj3VBN6jlKlsKinGUNP2Woc+x7381iiIfaJJQK73GOWD9ngljznnK45lsFobXzb9qJzxQ8SB8fDjxGg5PhzAkZR4YcxYSjJtdjlXe/r0M9lwAvh4mLmqeSxvJSkpszcF2b2HU9FEdlqaLtIn8z8oG73MRhRY8AAmJSwisC9Y8LlmovaJWcSbT3ZDRo80pjgJUQE3hOf99fBIqbxt7RfkleyVq5/95Q5Y20gK0dOS2AZP2jpg+q6RRo4xfjzfuAqJTLe4FRLttXAKcax73E/j+HjUB0tkZ43EG8tQ+caxHvgch0TtDOwWXBmKN/GxDoIz71UHKLhuOATldNEPTIlDBPXH1LVZWl/zvG6u5wTmEUQ9I+JbGZJHfo3Fz4WhhZuJnMVY2Tgs39QlWXVt3Mcj75PDi+opvCyTlEL5fWiEBJgyWjhlX0Mxj4Gvo+SdpWx2qetS76W0ZcevH3fQ4aemsBLhXGvuoMFdBd+I9sE1ObyJkifZYyVbKeBU4y/UBgsKtm1uW33ZrfV0inGMTXaz2P8lbwaS1gnJLqBJmT9YhGfh/CpXfCQDfli932VuZDcWFuMDlQqpCJm49q+qzmxFRNLlhicqxijaAd8OGGAPnnfD++Oru85eXetA6p7wTdKIHpYS5p9SLeXGoyyZShVeGsss2aoC2D7bHtpLdp3bqnqd6TXVY/dVUVlV+Nbh7LgSxNfISyV/FJz/eCKTvjAkglh8RS6LtldRWeXjrYa/37cLaY8jEszavWRgeKQQWTS7RtyufRv4pznscS0pcT+PTUGr4jQnOzeOVHCEZb5rgUOUF8OkWOEpHvc2akyY/ch2T9ET/vcc85zYgtPX19i0GFbYsqY8BrjJya+CQkolyBpBOqr/QGR+pRkpjgIBqqi+2SpwWiiKBV3Hy/PxJNf3REHL10AymDcjribpzBJuuE74DzmhLsafCA9oLtIvrV16cbBlpP9XBZV8xnBMEwqwWiAE26pgCLAyTEucPYBi+8jl/5NnMs4ps77ebwUJiJkK0THG8nwsAG+fMEglsE4rqTEf/n1kMzYBaq/75IyGKe1dsicmBuDJPZ2jfwLh60xYSA+vf0gqTe9zvuMr+SaLKSxz315LTzzvs/FaGMt5ziQ9DIVoXAYTRiH8P1oP5cYjFKreavg65aWlnpxRDyuvSUg8K1hOvb9GLbyMOVzJSVlGcBdbmj3yACEvyHCC7JtK1li31E8/Potxx8p67NaV3ClJiZlV+9AEL6yRQYsmhbPkv00sOU4enI/j5UjS1qMqV7xsH1Dd1iH9TsHr+F+Wjz86mN4GHl2k4R7abm/j24hS7i4pdRmh7/1ed95zDnBgGNErZGeKse6CfM4FIcvEA9JhIx52cT2yX3lLrKkHfo5EBiKmFGm5EtaVTC/cw6O4nyXGIwMRTUY1RbmOVwiH9MBk3WYAVIyrYGtJ1Xv6cHbhr9tSlQxsCjmGNlgpkDlNYbH0cCW489LLOxgPFWlAAoXlpwTJ190WULYSzzSx9HC5be65TjSRj+Pd1Ekwdo59OU8lvTy8Zev0r3eYGuD0QYLC+7fIBrCl0tgGih1kH1LkIElvmW5yzmxRO/vFRFf3i6ciobi4Mz67aB5CLwvRXqomWTlMRuudxZ4r0kP6i6DUQo5zOIcjc6Y4nSAZzGFUSLkeSli0+WS5SW14QKAvq65g3lsZC5uKVtOKmNK1/os89ppaAq/xB0v1OUwQIyZLEP3lZxOA1uOv17Lin9ZY+qXMYpAX43boUiKeGZE8F7JAuVV5JneB/d4Oi0d9iTzAQsB0mpGF6NCxRSMDy/YuM72luO4ZB67RgULYwasDoMKmG88Vfm5NdnaYORZdJgmsquTdWKXXiUpYhG5lsz0Xe879/sx5wQDHhvEGunHWEjWGpGHAvMLfeBjO8J8xuUT1jzwxPfywibeEhOKdSLtAesFNhROIvYNjOOszTBlMHJJKpMjBO2jh+NwUuK2nCuvw71rUnHd2qRS3GchExufSoE/sR5nH5cne4Yyok2cWrLzuL+F6N/vDDcarnQnYUkLibVwOrYp9iKD0vjALGRtWe51hrDas2Usnv5L3HJRzd7j3eMtZFAQQGaULKAGQiwOCX5jKMK5OEG/5PSvfvQnJjaPAY1vDQfZIyMCCS9D0pzO+bK2M1uM4655bM2WhChr01qrSgWR4WmdNo9v1au1hcEofJc4Np6kZIug0+9e8IFkecAqy/mAsk4xJzyH48OzDpG+2IgDNq8cY0rOhrWRDZBE7JxFYB9Z5OKQ553yHvkJ+p5lQTmTGMbWEc4wyZG8rEoCsvVEpxiOozJlMJocPFRD4ZGa8671ih974D48YqdU6vBZvDEW4CH2Eunldzb8gvqUuzIPl77D2knFU4RraYmg52Agfl/7OHxMsr+cig+VrBhjI2aQSrRgfKxpc2lffMMMX6F3hxzeI/pkIEyVglra9qmuWzv+c/Pboc8J2dxlZAg745Wz+fGaGHv8XNfkVez18UktWiAc49tIcbhiRFskd8E2ln4Ha8dxyTyGoeK9svCD+1irbGC+9yWUO0vfxaFCoQZ6Y6BaX14xQ/q7tN1jXreFwcgTbfNcK2BZ8MRrpNa2B6ihluxtU+HWJfoXARWOdoC0l5lbPMsqpDH8cSAPsZLmejpcljzjLq+R7CIK8cSGb2YDWP+9G+8joxJ8iYxVhXuw77tC0nf5knf5bJ4Z7tuxsE7WWqxqKA+MUE/h4f95Y3k5bchOfUKehwCEl4y/71eClTA6YfQw9HlQPROOS6LAOWUdLnmvumY7DQCa2wQyEahv+fkR8dQWkinc7j3NqF/t8J+FBkQqHL5wwaIesYEiAy85ngZqbTuebqvlh2qAISkKTHgb7amjcs4G47mernhlhKfRAKHhuHWxqQh783KoMsI7bezox6lMiJtrP08wW+sLMfwnNxoh9CE2Nn0yCWR68XTyGlSofWvNz7d3l17npW8qfKu6gbA1D0PJPQ04bJnHL2xUTQ5d4DjPbvyr6Hq+qRR2VA3U2nZU9V5946InvKdTVY9SASJMoglgg6KEk5G5czUYz/V0BdfAg8aFKyTt5F0yrQH4KuSqjEVZnFzhpxShRrl1OuAAAA++SURBVHhUhisy+ZLTaOAuvc77vCEaIWEoi6Vwa8luDcDoSY4ztxycl+D6drdaV+yrgVrb9tXY7V0vSmptm6swB7KUkVQJMKKCk3KuBqMOn+Pp6j1a2Ty4qI+8ve/voDfOkpIf1RINDmpkxU1ZJ/RQstUVj77ZW+/a67xE8U7e39+S+GRGlizXgCoQwPHC05g0CvKxXHdbXllr25bavL62sva1YgyfN/J6oFwiCPDKooTgObNyzgbjXMfv4nQFkydD3OLoZI2Xq2S3BgDmfbi8i7yyfeLB7rvXX4FjUELArsLr659ULSzRwF17nbOPTtII62WKy4ws2U8DuF0Z2u/U4Cj73V1Xb6GBWtu20OJ1tpEJYPl2Mr9F+yT0WIN5HpUMhvWH715UqOFSDUZKOPXpygbzmBZaxfpeslwDXxURvLM7Xd7Lm1x8pRAaXCXOLjQZiFdL7lYDd+11Vl7MiVv2uMIEJftrAPYTBhRYHs1Yyek1UGvb6XV+SU8UCZDEglIH/ljSmm8GtZrqfS+NiC/Yh83kkg3GU56uVEzgrsXeT9El+2lApjIjQVIMrk7S1/D0/4C2qBEUQB8T1ABCYK6Bh9wnSzPrac6VV3uX1n4+myc0yc+H/ZHQIZyZwoPKe1qyTAN36XUWQlXV5sldZuCyXtdVvQaypBpMN7qmDEvbhCTEpOziuYQhhaH61MaoUFreTwO1tu2nr7p6hQYu2WA81ekKCzpqDi7cMhYP+9jUVmV0437ElUiQDqum4xQki5pMcUAZa6n+SlSiM5ojjx/rYZbqmvOGaNdpzGb3Nq0R5OZZ6aFvl9EqscPJTX1Z4GIbX8lyDdyF19nYMk6eUsbi8oGauBImGPUO6TnwZJzzZCjcAGwPisK7MRaVcfAzN83tDygs5EFjUmvbQWqrmw7RwCUbjN537HSlmoMqDimItsXqp4iJpZ7D48Ax8Vr2KeWoJWTXMhZf1bXJSyFD+hsPUfoN3mOD/qy2IcCC9vVYbS68hYC3/o4cfZh5yUMJuCusnYz7+6hRjV14ta9vBufcvVkZxDUfEhHPm7lYmShGZtXf3mc0Hrh2zOs8bAWfp3mGnSBlrqayOdzXeWWoqOxCsl42T/cXdu2peMPo/9L9X+Gm76DnxH6OZZlbg63FRNjrSQNtIT1GxYWmTDWVLUnHb2lgam27pdG+43e9dINx7HSVNWOFGHPz8DcTayjAn4xFCSzA231JLUz9SiQKQyup1osFEPs8z1nJPQ0w/njlZMnaCIRqCZZ5ZRbJmzZqol5vKiIksfLQC0jHSJXhpMbGMNvxLb9lK3cknIwjLgnDkz7FASPrZk+NW9ZP9/uLGp/j1LX4rbzvexYWbu9pMOZ1HjYizC+7XqkqHmCitmt+S/31vNCf3rxdxlA1lDdphxRrgYo2IAnmbi88YcLTPGYl92tAZSAlxMAwVMbxj/BzX4NXeBrcpBdZ6A6CatnjQDXv0sA0ruancDa+1mMR+1/LeNbadi0jeeHvcekG49zpSvgJnoa8upXx6T1bwiW412QT8VoIl6bYpNRVZiiOZVC+QwtplcH4gMYYipKC+lKKNhWbM3A8w5sBTmwkMrV6sbHbXGAGeYLVQUbb8YGtPq42YJymxPWMO+HtFGUbeQuf02qDOhDwEvM2z4m+KGlobrymEZmOXY+Lk/dLKUTvf6v1e3eoc/LnOa/z8CZGRfIkfkVEPH7moepCK3flep4vBz7hb0DvDKH2t6Pv4t0qg/GhSvVd85yjD+s9vKIqxs66iXmAoOeACx0KMnRRGsIb/4Q2x40DQ1RI+liE/od+l+d2X61t5zYiN9yfSzAY15yuhDxQ8JA+Q9diCHfG8LNR8Dz1wnuohuWcMCjLYHzAq2cToS+ehGc0biebDKNdWIrHT7kxRiTdj8EDbCxZbcPm87nNKyTdn1dpSoSpGRG8msoCSk7ieeYR4Vn2u77xaqjMw/jYJVn+0XXuS09pfx/PIg+j2uoSeErGNXCo17lvzeGOx9A3Jdlobm6qROJ78e+PaHWBfQdz4vstg/GehsBGHLYTWyyj/BPa/LJmCjc79AnzE9chyB+KA6CDYHp6zZNP6Wq/O8iX1NpW38CFaODcDca1pysLGf4hIvsWOBtGkefLaVcSReEQD/9YhQGFliSw8N7yNDDEe3nziPiW5pHguWNAjInQlc2FB5K8vhmMNvspYmDGHGPfxsTI51Ea0uYYYyUDyS7vVPZLLXGhSsKb5b6hMIyVJJTMk16UwzV5fXeu9ToPNcLbn95hXsEpaivz3byXhDT8Fq9Py8d5I0YdY5soUiDi0s9B2FLVXswtAgMKKz4moCYgJ8SczlKhu8qVHefNLqfVWtsuZ6xupqfnbDBu5TnKDcSgymw1ES2Gj2sGyc0M9hFelA6zULmQNC/uUHxjDEVehl0evj5hCURAQtJcOcHntgx2z4RxzALqfR8cOr69/YEHU5h6l/BSPr9dBC835uH0LrKk55Iwdj3nWn/fyuvc68d4ZCWCKUyyAwzPF4Pm4VW686DPy/ro4MbLKGEIzGOsDrtsc2XFhPMlrE3NU79pLw+Kc9RWB3X4Sm+qte1KB/aSX+tcDcYtT1dC0kLTecK1mciUTIPgksfvrvvO88arSFDRjNWV5WF8bbvmMxqecarfya2Zv8+V8xOeVCw9s6wlOPUY1WwDSXNmxdrgJD/sEp6sTHR6ZTMK+3vwzjFobYS81iX3NLCl17nXK9qVTFiRuS6DfSg83SACMHKM+ZL9NUCvWfUBhpi3fUzMdQc6hzFJMVNingpXZ8gfXEQSUsm8zmptqy/k7DRwrgbj1qernky2J48+uwG5sA4xBBmEwv1vMdH3rL/t57FsyryNJ4Nhz1PBICOZuDDWtM0qDdQvapyOY9epRiFpRkjNQcRCvEtscpJY0nMiXN7TLX14S8LhrUzc5a42b+X3rb3OqTcUObClBDcmDOlQMslCooZQasn+GnC4csgibx0R3zPShKpJP9z+jh6J13BKHMxRYrk+4SYy3lFclYxroNa2+jLOUgPnaDAew3MEsG0TIcKjJmQueGc5MBfSKZmnQs1zdZpT96iL0BiNUWhIYoCFNDa8jGpeMkQJnFRmY/Zq6UPNMrATJzVUXda8ZYzKlF4qPZSh93SaM/BaPIu8quk9XdrutV+3tdc59UXvr2tGB0+yajHDuuSI9dG3PLLG5eDPrA81i8aMhaP7SMDU/NQB8xIlVpJ4y14n5QGeH55a2w7+fOvGY2rgHA3GrU9XCJth29A3pOeqPEPbfFUWfsaUMK9w71AY/8K2jD9YRyTXQ5Gwoh3ePDQbPIB48VR9IQxJfx+KUoGMT89A2yPcPRRGXRI3q9aTG9aSt+8PGRJceCqJd0ARIotbP0seqoEtvc5D3X5NROBYJEOy6Ee3g4bwZ2b31tjsrwFJXNgfeNgdBsfkixsDgblqbo9BQXgV4ZrhkiWHoaFCUSZhiWBQUOqz5H4N1NpWX8VZauAcDcYtT1c42F7SqDWQ+sKlwVih6LBwDfkAz3KQzrhTmU2JpBt+bCg9CTacUyaf5HWocMAFEDMzCjPDGfUOguAs0SesbRyHIvsaXQ9jjlE3FGHJZzYqFm3J0lwqsmwRfpM0iHlc8EMqb6gizVi4bmn713rdVl7nMf301EuMGtRWKVk55omNk/Na9Xvs93pMR0LvEDf04gorYyQwF1TBAh8aivlqzhgTiYYpPQ7V3EXRUzKugVrb6ss4Ow2co8G41emKgSKTFVaG98spOE/GBgKXGLqIksM1IBOVASXxRBjQRpKCO48hqQQYHJNEo14Y7sqx2aB4iyyQvdj4M+t5CrMmxGx8YdtgIHvqD89Vc5pRysu8b+k34fMkFcbviKg4vY5TBurhmryeO7fwOk9pow+FOkAwTAgvF08y76ZxnyoDej1aPt6b2BNe1hJT3icivqx7lN8kwTD8HP7wKoKa9OLgCHuMLN39/ZwUDQAbMEYE0T96npL7NVBrW30VZ6eBczQYKWnt6YrxArumUgcvo6ofhJeJ54oHy8mZl1F5qpLDNWATF6KlbwkPvHj0+sKGGeRpkOwyrBUrhKwqB0+eUONQGJQ8wjjeiPZg2IS5MvnB34W8GHJgBsJpNiiLLQMRebffeUIPkcwEdS8+Rm0i8WacVjmzcY2u9TrPjRP4gqiAdYsn00HFeKvywziRrOFQWLJOAxLYGI0O78L7OC8luvDYm7NCy7CL/QHRE0FArAOMQtGFMaod8yjLCIooMBgZ+pmZva7n13V3rW3XNZ4X/zbnajCuOV3xeiHjtpHA10mk6OUFHV+gsnEyXkvWaYCHR+m+t2/ZxGmYMwpViRgSbyfpNTA8Q3BKeCCGNDg2pWGFCOMs6SUztfEAou9gSPIyHyo8nDydhCfFBlp1o+e1ucbrvGSceLSzBCRPVRo36HRwYk6RvC9pu665pwEhZwkrDnSiMw7hoDzms+jMMFQNHw5vLJkQpdFUUqE9B2k3zGnKsH58jcM9DdTaVl/D2WjgXA1GCjrkdKU+qYxamwhMHA/jUODPGBxClYwAiRA2GeHpoRfsbAbqQjrie7JxqLE83FBO8QpCXkLHvMZzhN9L+9Jz0rlnipx8aXu3ct2hXucl+pFshB+QwMc58JnTQqGgCyXba0Dyi7nNu1sG+fb6XdJirW1LtFTXHFUD52wwevF9TlcmlBrETrf+edWM5ngWZdamIGG26ZSUBnoN8JjCYRIlCBkld2EIX+Ko7Ot1XvqOqgmJEpD0+hamdKn26rrSwAMaqLWtvoS9NXDuBmO+0Nanq70VVTdcrQbQ5MAlZjWg/kXh4uDjeFYA/JOi52qVcYQX29rrDNPaHwaFqJF498TqR3iNarI0cHEaqLXt4obsvDt8KQbjeWuxenepGkiKpa9uSRP5HkihJdGgDeFRxBkHzF9y9xowZjgCJWFIZuP1ZTSWlAZKA/c0UGtbfQ2ba6AMxs1VWg1ekAZwKcqaR8MiAxp9joQdtW4luMgCfWwjhL6g17r4rqr/rvqP7PuhqOSCSgmujgd47JqLV0C9QGlgpQZqbVupwLr9fg2UwVhfxS1r4O0i4nkt65ZXUaKMrHqhZ7Qi+CNlhpacVgM4Fh/Van9nkgUqrKe0DF21vSW8FBXLacelnnY5Gqi17XLG6mJ6WgbjxQxVdbQ0cDMagCmV7a7MHwqWRzSvLzJ1iS5P20HHdDOKqhctDZQGSgOn0kAZjKfSdD2nNFAaWKIBUADEzrI4EXODC/DyIo9WRUa5OYTqJaWB0kBpoDRwQg2UwXhCZdejSgOlgdJAaaA0UBooDVyiBspgvMRRqz6XBkoDpYHSQGmgNFAaOKEGymA8obLrUaWB0kBpoDRQGigNlAYuUQM/DoG3tvZe2XlKAAAAAElFTkSuQmCC" width="326" height="40" /></span></div><div class = "S3"><span class = "S2">First we introduce auxiliary unknowns </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="48.5" height="38" /></span><span class = "S2"> , </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="51.5" height="38" /></span><span class = "S2"> with total differentials</span></div><div class = "S4"><span style="vertical-align:-18px"><img src="data:image/png;base64,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" 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" width="161.5" height="40" /></span></div><div class = "S3"><span class = "S2">Note that this corresponds to converting the second-order equation into a system of first-order equations. Next we want to rewrite the original equation in terms of total derivatives </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="54.5" height="36" /></span><span class = "S2"> . We try to expand</span></div><div class = "S4"><span style="vertical-align:-18px"><img src="data:image/png;base64,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" width="371.5" height="41" /></span></div><div class = "S3"><span class = "S2">and compare it to the original equation. We obtain </span></div><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="121" height="36" /></span></div><div class = "S3"><span class = "S2">which after multiplication by </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="23" height="36" /></span><span class = "S2"> or </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="23" height="36" /></span><span class = "S2"> , leads to</span></div><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="172.5" height="44" /></span></div><div class = "S3"><span class = "S2">thus giving us the criteria </span></div><ul class = "S6"><li class = "S7"><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="120" height="20" /></span><span class = "S0"> no real characteristics </span><span style="font-family:Symbol, 'DejaVu Serif', serif; font-style: normal; font-weight: normal">&#8594;</span><span class = "S0"> elliptic</span></li><li class = "S7"><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPAAAAAoCAYAAADAOHfQAAALEklEQVR4Xu2cBYzuuhFGzyszMzOozMzMqILaV2ZmlemVmZmZGV+ZmRlVZmZudaTxlRsFnH+z+ZO7Hulq724cx5546JuZHEClyoHKgdVy4IDVrrwuvHKgcoAqwPUQVA6smANVgFf88urSKwfmFGCfdWXgesD5gOMBfwY+BzwLeHl9HZUDlQPjODCnAD8QeADwTuCewDeBkwEPD8F+OnCbccuvoysH9jYH5hTgRwA3AE4dljdx/jDA14BTABcAPrq3X0ndfeVAOQfmFODbAccH7tOyvGcAtwwL/eDy5deRlQN7mwNzCnAfpx8N3A3QzX7Q3n4ldfeVA+Uc6BPgKwFvKpzqv8AfgR8BHwGeCnyh8F6HvTHi4IsD7xtx3xKGutcUu78SuM7EizLseGHMeT/gITuc/wiA7/YSwHmA4wLHAP4K/AL4EvBJ4G3AF3f4rCXcLlh6a+Cqgbl45n8AvAV4Wvx/CevcaA19AuwLvlrMeo1Ajf311fGi0wMPB5wIuBDg4ZD+BdwEeHHBqjw8Pwk0WnR6TXQx4D2wL58+tQAfMcC+EwRTXgtcc0MGHR64K3BH4FihcFW28v4fwLGBc8QhT4/4DvDcyBL8esPnbvO2ywAvif1+JYT238BlgbMDfwBuCrxmm4vcybNLXeifA8eJBx0T+E3LQ48ejNCKSn8Jodcy99FzgOsGQ7+xk83MfK/CpbU6efbcqQVYa5tjBgrUqTbY51kiTXf6sDjO+aoQ3OZ0lwSe3RBk/6aiWhOdG3g/oOJyP7cC/pNtIIVtGpvLA+9a0+bSWksE+IThGnuPrsdJezbqQfl8dl0NeHDP+AOB54V789aVMfApwG3jkFw01j6lAMvnr4ci1EuRDFWOGtazlF16Um+Og2x4cnXgdwM3nzLc5+RR+fzflj5wAeMOGcpVhWW68kwtyuoQwKfCcBj6nSbCiAUsv3wJJQJ8xTgAzmpMfJWe6bXOv8quO7YrjtaNeT1wM+Cl5UtexMiLRKz+mQDejKekKQVYC2noYryaK7cLAx8q5IJhjXl3rZAFM6bpjHVLKGUGvgso0GuiHDe4PaCybaN8nOHF49a0SddaIsD3z5BhEWKR4i46axyUdN3f28AsLfPrgFusUHh1nQV3jPuNGf359okFWMH7YMSeptfyEOYOwJMLDpohjS6+HtTfgTMAuuClZFjzssA8rlV600LG6fVdKtZyEuCHHes6WoSDysFn430uZAtlyygRYK2kCJ7kTxHjLjIVZGwh/SwOTx53+PdLh+UVGXxRNpEHTFfnFWVL39oohcectortoABEphRgXbtPR2ytW/dL4B2ASk8y5BB4GaLnAzeKQVoWLcwYOlu8yxcEEDTm3m2OVcEaIhwqzqC1B31kmHLaGKAy/vE2Fw/o/gu0FVGJAH8fUItJxmXGwW10ZECk78Rx8U7AExsD1YpviNjRg5GTh013+oJFK9/OIN1XgRG9CkGSf+6CACucAnt3AR4f27Tc1PJTSVdYBLWPrGoz9vMwqEB9b8Z5e4F8L5+IjX44siN9+zb8uUIMEMxKyngbvBJrUFlbzJTCst51DAmwE6b0gcizMW4bGSOZqzTGkp4ZuTdBl0SCKS7KedpiONFcBWKpAiygo+usMJwrA+uM5aeywCrBbwVgdObgh/zThTW+lnSHHSevuuhJgLGfpDuZrPc2DuXcz8zj2hJMwrNqKCdtOw7WAzAHr2tvNufjQ8wbEmCF7t0xiS5xDjZ5r4iorq9az84i65itedZKNcl8nJ1IfWRecqkCrDdh/KnbrPucaEoBfiRwj0hr5JbA+nEtaiLR/r4iC2M+D4NUGjMPnZW1XNfzS56LnszNBxb+2PB2HGbazmKZbZLelQZOsFGD2JtaHRLguwOPKtiNPvt7w/zrGrfliQumWewQQaUPAF8GztlISUwlwHoxXw2FmVy6xBDf0+/D8vo3w41UndVkmgrVdSbSkgtmbYvuHBVfmz7fVN2YIhIFMNXTiz4nT6Tr+Q8F7h0XVdIqgG2TmQdDTRWxxU0/7VrQkACLQopGSlpPf89JwMCWQMEtiwNMVyi8VgutrSSyi0e6zsa87vO8gKmjnKYSYMFCBfeMDWubnqVWTt7JEwAFo41uCCR8QS1+pEYBw9yH0yon02GbkpjKmPg9z5qUCLAe1X1jcYYeVqotgfScVCh6Wlb8tRrFIQG2ze90sRuT4iJ2XWTZpakhSY1pxdBQwcASGDW0BoXFlyqQlDT11AJsvGOlUx9anMe1hii+1DbKD7CWVwu8l0jFlvK5JS70YzKEXmuchHkJPEthm0IsAGyt+v9RnwBrXa0VNa3xp4h3mymhfDLHWa1zlPjj9Xc5x2udtWDOJmQ+tCStosXTdTYOMa0igNSknVpgkWKRZYvujXV1ldvoxpFC8pp8TtVZzbGpQsy/G0eLrO4lyj2QsSCWaVBj4jGUsKAx95SO9WzYe6Cy1niKSVm7vo/6BPj80Vnk4BI43nGmmFIaaRNmlG7Mcaa3ulDxoXkUGOPaIRJUkw8CeabI2khUOuXJBZpy8MnYRWCqj8yH2xUzlkTtv9dyUx7TGUelhpSx8691vB1WCb0tObd5GskQxi6sMWQzT2l125h528ZqeASM97nTfQJssUKq+CmJDUyc28Bw6Hiydc4iz2smwSBBoU1JoTem7SLTBaaNfCEl+UfjovTOFEwFtEn5e1PxpIqk0j34hRT/SWYW8lRg6RzbHDd3IYfn3WaJ3SKrGc06SA8LF3/fO+kTYNvIbAmUdEvyqqm2xeYW2+t9RR+7tdltzLsTF9p0h6inLlJb6q25n29ndcldZa2i2Y6TRDFTEU4JbzyMAnbiHTaliLgXVwV1PED8wLz5pqTSGpvVsLPIDiqptJSypEBm0z1sep8VYh8DLItt7QXvE2BrQ437JEsc89RE24LyhLj54FTUseni13LfpgLsyxFkslGh1M3NEd2+xhJdx8R/XS67bkootdjZH2z8X3pf39xzo9CuJY+DS5sZdjvkK+F/Psb+bEMBq+o619YlwLpQ9vH6U/9esKhPE5teMc2hGy3QZbBdYlHGbmqJ4zcVYAVXK6GLnizm0P7yHGdfa6dul8Ln+xCE830MWVLTgBYy6J7peTVLXYfWtqTrY9sJrX8WQJwrlh3ilXG1WQm9Wt+LrnMrdQmwllcLLKkFur6UIaOuHZ/QMZ6TrCRKDQ1DC90frm8iwOmesU0Gzc8c9fXp2sXkp3p9xyoLG9rb8qlW/nhALLf0AFtWuHbswnO11oZ+sznWWyS50oPoxCFyARY9Tklsu2A8LJLVQU2A5bDxhUk1ROr20CJYgO9nX/Z3kk8JbbfAIxUqmG7Ki9DNMVqCmsh41xJHXWZdI3EGc+WWqBqDdZG10B5IUffUYeRY73N+U2ptbZsW1AhEmqLSMzKeEsnUu/KzOrZDpi982Dfs+vpy/Wt7rypKlZF8E1D0wwZ6IpeLhhD5YPOIqZqlkMr0XiFHCnGv55QLsJVWpZpXTe3mbfY2ySz0rpAbO+0Fst3Pwz9ETexAYbV+vEmGIKmDpm3OvAqr7boeU/4llHyMlVhmBFKVl59GSjl7hdVUmR8P2B8+YNfGG5WXHx3s+qid6cilkEU3KmJ7wfWI/ja0sKFKrKH76/XKgcqBaTlgN5WeQtEnjKoAT8v8OlvlwKwcqAI8K7vrwyoHpuVAFeBp+VlnqxyYlQNVgGdld31Y5cC0HKgCPC0/62yVA7NyoArwrOyuD6scmJYDVYCn5WedrXJgVg5UAZ6V3fVhlQPTcuB/HzUsRw6H1BAAAAAASUVORK5CYII=" width="120" height="20" /></span><span class = "S0"> one real characteristic </span><span style="font-family:Symbol, 'DejaVu Serif', serif; font-style: normal; font-weight: normal">&#8594;</span><span class = "S0"> parabolic</span></li><li class = "S7"><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="120" height="20" /></span><span class = "S0"> two real characteristics </span><span style="font-family:Symbol, 'DejaVu Serif', serif; font-style: normal; font-weight: normal">&#8594;</span><span class = "S0"> hyperbolic</span></li></ul><h2 class = "S5"><span class = "S2">Systems of first order equations with two independent variables</span></h2><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXEAAABQCAYAAADvAEWDAAAeWklEQVR4Xu3dBfwEW1UH8IPd2IGYKHZgd6CCnVhgYYuBgYGFitiNHajYgYGF2I2K3YWKgd2KrZ+v7x653Dczu7M7s//Z3Xs+n/d57/13dvbOuXfOPfec3/md20SXroGuga6BroGz1cBtznbkfeBdA10DXQNdA9GNeF8EXQNdA10DZ6yBbsTPePL60LsGuga6BroR72uga6BroGvgjDXQjfgZT14fetdA10DXwKUa8WeLiDtHxHNHxBNFxKMi4rsj4q/7lF+EBqzbF4qIV46I20fEP0fEb0bE90bEv13EE/aH6BrYUwOXaMTvHRGfVT3/X0bEk0fEk0TEl0bEh0bEP+2pn37Z9jRgzX5zRLx5Gdp/R8SfR8QzlHn96Ij4/Ij4n+0NvY+oa2B5DVyiEX/SiPjiiPjxiPimiPi7iP9D4bxGRHxBRPxXRLxmRPzF8ursdzyRBp4nIj45Ih4cEd9XvG/zfveI+Oxy6rpHmesTDan/TNfAzWjgEo34lCafKSJ+rhjwV4qIf78ZtfdfXVEDQiw/XDbs91vxd/qtuwY2oYFrM+KU/l7luP3hEfGJm5iFPoilNfANEfFWEfEqEfETS9+8369rYEsauEYj/rQR8Zjihd+uJMW2NCd9LMdr4C4lyfkDEfHax9+u36FrYLsauEYjbja+MyLeICLerSQ7tztDfWSHaOAJSi7kqSPijhHxO4fcpH+na+AcNHCtRhx6QVhF4tOxO+WjIuLjmolzNH+bkcm8W7mHj98pIr7yHCb9Ssb46wWG+N4lfOaxnzgi/jEiJEFrAVcEURySX42IF4mIP4gICdUuXQOb0sC1GvGPjIj7R8SfFJxxTso7RsTLlpf19csf7xsRnzQyaw+ICLF18pIR8Uubmt3rHsz3R8RrRcTXRMTbFVVIbN+v/PdbR8QzRsS/RASPHVSxlScrsMUnjIhvi4g3u26V9qffogau1Yi/e0R8UcESw4//ZzM5HxgRn17+9noR8bCRyfuuiGDsoVyeKiL+Y4uTfKVj+tqIeNuI+NGIePVGB9b930bEbSPipyPiFUZ0ZEP/mfIZ49+e0q5Utf2xt6SBSzfiPK2XLsfnny0JTfr/gIj4jDIRzxwRCoJq+arKe1P9+Wcjk/anEeHzX4iIl9rSxF7JWKzf5yqnIFWbP1UVcn17RLxxRAirCIfUcoeI+N3yB5v5e47oKzd7H79JRDz0SvTaH/OMNHCpRpzxVp3pxUtRwfcVEXGfiHj/iBD/Jk9fvLJ62n4tIl64VAI+68h8Mv4qBcmDIuJdzmjeL2GoLx4RX1fmKZ9HaORjIuIzI+IHI+JVI+KXI+Ilmgd+y4j4xvI3uZEvHFFI5k58/JwR8UeXoLj+DJelgUs04jwz2OBnLxV7H1ISjuKad42IDyuwQkdlhh23Sh0PVaIv+eV6XByvOzLl7pVhFkUlD7yspbHpp4Esekg5YdlI3yMixMCdilRq+tz8g5D+SKnWrR/oEyJCroO8YkQ8YuRpefZCLTh3OAZdugY2p4FLM+KgZV68lytxbvwa39Fo/TlKnJOHPfRyemndgyjtZvSHBAdLJjx5fMr8u6yvAYZZiEQ8+48L0VkLIYQNV45PGHsoolpsvjZhm7ekJg++FWvpHyLiKSOi483Xn9f+Cwdq4NKM+FsUciTqEE6BA2/FMzPeTzfipWVFp+9JjH39iG4d5UEPefMMCu99SD4oIpAy8ehzczhwuvrXyonnfYomoE6gT1oRakmk0MeWEEt9De9dOOy3IuIFR7T6AhXsUJJbGG5I3EeZv42kDt/1yeoaOIkGLs2I87qS3e75IuL3BrTIE390+TuyJPHxWr4kIt61/GEKP/wbxQD4Db81JtAPL1OO7Yl0OMnkXuCPCHFJMgtt/H4p5GmRRR777Qs5lv8GCwQPTBFm4cGTqRoAEMTcwN3vq0f0Cb2E5lgB2RtdoM77I21cA5dmxBlnRloCSiJqSMRLvXBEoY+Cn1oeWRAt/1qO0kP4YaX7f1PYEdGiSpSNCaMBfsjr63KcBp632pghiN5h5HafWjxnpyThlxpd9IZViG2KP0dyNDf4F40Iye4h8Q4JwUEoWTNdugZOqoFLM+IMqzDJVAwzk1peOMUfNbe4OKj/l9wcQjXk5NToBoVDin66rK+BOkwCXfTxIz/5k+XkI08hX1GLHEcSnzm1fevIPbJS0zqxCaMw7tI1sDkNXJoR/7HCXDdWXec47hjOWx/yoGvYoMQYIqUhgVrJz3j2jtOt2ARerPxxl7e+uYWx0QHVFZS8ZOGwVp4/In67/FHs/POaC2rYINpaBr8ViJX8u/oCifJWbADCdylT3vpG1dmHdQkauDQj/ikR8cERMfbiSS5+T5m4oTJ5nnk2ixiCpvmqCk2VminCJYp+WoE/V2zy5RHxERHhBNDleA3IK4CHCpmAj7YCMQQ5ZE7kKh7bXMCo36v8TXMQSclabPScAYacyJEo+mnFhsJDVxGqDSCUS/fWj5/ffoeZGrg0I65Fl9ilxJcXKxNY1AIqxrhLVirugEJphT6EZMS8VQBCKOBXSQFd0/rNCyvmzuA/y4TOlWk79jP8uXnMnKJ+eaMB4RGGE2GVgqy6DZu5FZtGcDWGLKqrddvEtnnVFcrcahpCGHwdoYbE7wi//XxEvHyfqa6Bm9DApRlxOhTmUI3nOAxyyBPDPodhkAEQapHQHOM5QYwlzk3+qiRBvag8d/F28XCbgU2BtwdqSD63MN3V8wijLpE2Vbp/E/N+7r8pru1kozrTfDHk5haahK597lQ2JEJmkEVOSsSJCxyRA2CTZrDFwbMGQCjsD0uIhoGvBaWDRPhU6f6567qPf+MauEQjTuVi3rC9oH26nztWe3F5Xkrkp5roSm7y1lDL+h5yK14fQiUvKw/fvVpRPJRl+PmZk4DjOcPSZVkN6NojsZkoJJW6qjYZd4Z5SnjwjP+rRcTTFPSKsMrnlPZ9NgMQw1qGTm+gqMItuFesjS5dAyfXwDFGnDG8ZxnxVisWPZ9wh0Ic4ZFTSsbXhVGS1vaUv7/rtxivbCK961qfq2pEFKZHKQ6athJ2n3uscY1N0gZqbKfumZrxdaGUc6oB4FQoTELVK/lurVoPf1/Ch04mkD0S9nVIco356/d8fA1wMHDga+zOMUELAiLLkWRLOBqP54QeasQdI4UU8vvvW8IJfUIep4FsEcYzlNjcmvBgMzEI52xOiTCUuHKKReQlFyOuycB6COEWXUGuiKW3CdStzbfxOKHK0eDNR8EsHKgYzQnSPEvS25Ay1OQ7eIhUP6uM7VTL684qDibsqhwTJ0MbqUiCgjLv32Bh4aFGPKF8+UhfVlU5rvuY53P35FYRQxdX3bLIIWTBkjzCtwwM1ktuQ4L+SRFDhsm/RkluFR1/wAu3LvJANl5Je5uPvIH3uBXG/Z1LOPIpyoeqYm1UvZhpvVmuUW/mBm9Tio5U+O8VJmb04/8/PMSIZzkyFEfu2I7Y4s9dHqeBjKuOlf9vSVd4P5I6QFUkLP2Q8BB4bZKARNFMdjba0vOcYiw4V+RGlOMry9+yOAlmYZRckY14Ki/kWfACZdJeeEViv8t6GshKcacjYZSWTgKsFlLuVnDmuUZcJaPYjO+Bz0nqEC4/zOwQj8V6j73tOyuzF37g+ex6YW7ySXhYYqHm1L+Nd0p+pfI8labrgnSNAsIo2Y3gLBuMbFEPxvdpZWBzQ2DJD9T58ted2ZrPae4c/X9Me98hZiPhu0fELxZK0PyukmgveJdbNjQGUUyrbQ22Nf1AaCSaY6zAKcdc4+j9baxqcmvPuMZ4srDszhHxQ2v8wAL3lOuwBp2gGGTe9JzkL4Oi0KluNr3AsPotGg2wp8nG6QSEmG1vmeOJS3rwLhlv8C4xQaiPjJshI0JK1OUW6BqDOEVhOldPipjEJDMBOff7Y9ffuxQw+VwhE3jlmAiZSWinaHuG2/saJTsHSfpC+RwrwpReZDkJreWOFXFUawYNAVE9PBdRBGaL7teaWMtBy+Kt+nmF7MY6ainky3aKUFK3ihEfq7gTfb+mh5j6yZ3V3nOMOANtx5CNFwMn2fnEf1/z0drO6ZSSnB2Or46xS0LPhLEY8aVjk4qgkg0QauHBEysKWdSbls+vqfu78nw0sxk6kgt6TEnqLgUfzRh0S517qE0xl4wcsS5VH29RoC7YFQnV5P+f6qj1OhHx8PIg53wSlAzP3q7mXp5JE5LWERayTm78wfnb14gzRgw2HpC6l2RNJuRI6Wh5bcLjcSKRI+DJwndCo5gMbcOWkrWMeE3UpRel/28FhhhXSXKIgKUxXpLb1yCaekBsCI3ZSFV1Wuv4VYYKvw7RydJGvJ7XKcrdQ8a6xnc0Gk/ncKqjlqRsVuOaDxQM5yxsMOMtBDtFujf6jPsYcdcw4Hgp7thUJdo5sxTZC52ohXNW6tyxK78HwEeuJSmoi7pYon+WJERaw4gn9wf4IFHRWienhcqgVRx5eUo4QmCGQUqHeNbn6u5crkedwJArxPDCgeb5/yU54pc04pBGdcu6c8hXcQ6tLTLVUUsy2efAAmgw5J7OWeoOUjYn0ORZso8RFxPjVSoM4Y3V0sZIlaTjmeiyvAbWMOJgS/tWGjLgjrFOYxk2Wv4pr/eOSxpxiUhcPgT/D76YLSOkjLNml5zqqCUH4/NHRcQdFlwu4v7HkJjRubaPc6WGck5tXqP33WXEeZm8DSXXYjhtZhsdp1BCenLipUskZeYq4hquX8OIC48k58cQ5SrvW4m20MH9KvyqYgRJ2y7LaWBJI26jlZQkU/Hl5UZ//J2ykQdbA/Y6dNIDcWZvoG0UpEkCLyVCoMfcD0zwEIoCoaOsnFZ7MPt0t8uIJ5XqVGa7jr1hldOYtsvyGljDiIvtZnIFNS+SpzGxSMV/bezkEuKRy8/S4Xdc0ohDzkjGkgdGhHLuLUuNdJNvAY0cEp7yI8oHU52dtvys7dicbiVrcTvJPc0OU04ZcdwaDIf4310ntFKjG6YQC0pGTY7dZi4Z1T4dxXENbBmTvQuDTcVizeLUQ2IjFWOvG1LU16HLTYO87yL2wmTXGv+u4YND90hqXZ+JXyaawP+Lm/OOULge4qWbO97Q1Pe3HhIYajJR61EXKEfmIRGK1GlIok6P2CGBHEpkxtQcZ2s514ixjtHy7rtO6uuOeY/dxxoRxqvtgPBIQlWnil0ABdLRgBbKXrntc+yzlg559jW+Ay4JNukkYv5bAWFl2+TakJbdSqaMONgcvoU5Ih5uMbbieKT234DFsRyZ5sg+HcUvwYgzxOnpztGPa8XjLIZ9xZHU0dQRVTJTdlzl7ZQwIlli3nZ3z2PhoRWM+f37TGwC527EPVubV9p3vly3L9Ec0qpsarEkDO/Y9zi/b63i+E87cI9CX+AZp06E9clxKnyxz1qao/e1rq0rNeUEtBNsJW0fB4ojdysZM+KKeXjgjmW7QP6KgO5W3VnGeKj4AeUlSsUE6s9RTO8ofsupaEmcuBxHzu1UU+h6nupjekt6ZhOA3tg3UdrOv83LmJwOrlGWDKfU8d1DGnnb2G3yjvatw3XMe2xezbGOWNkG0d+yrsJ/23yg4YYEu6Y6CclaHuqYnMtaYpQzh4ib3jvVSto+wIJBJ2vIiItPOVbzqFUV7cIC3745/u06Ul7jC7rEMy9txHnUWdgjEQYHPSXJpJYnhV2FQUs88zXdY0kjXsP1IMuymGsffTopq/QU1oNw4f2vLQ+rQrYKqZzaW9EXAPkT+6T5hzjykqJyWZjnUJF32GUr23sDC8gjEpXYDPVsGTLiXmY7wpxjmJ0xMeKq2lRvEi88XC3Fk7nJiN5R/HFTurQRR9qUJfb7HNOT8MmIhH0wrVm02ZjY3w8hQoOxT0Oxy8OavcDP6AtLGnGGEEWu8IU5ut0eoTKqslELS/KGMVneqeCwj32PjaeG3wkbCB+kJGwQXzlE1JDct2o2PtYk+5i1dBPoFDlEcW7P7SRbo//qpu70Mdp4pzXiJh0GmOF17NmXBB6ndFZr8u54aYTxNoGSK+KBc/kbekfx9Yx4nUOQUJFYGRNzyEtguAkPAnKJWHyy6vIhwjJzuV2sOTwZ1t1BFWtnZKinhrqkEfc7NVYcDS0HakoUdqGe9Y4Kh6o8znLvY99joRlhVtTFwgbteuP5C8VJ3DsBtEVyQrZCf+5BxNAV/bRybmvJOwNAMkT1Sw+e56EFEKKQkF2+lbRGPL2qucYWGiF5JYbiq8m5wQg8euZLt0RHcTh23oXKNQZDPI5RkgDaerIs1bWkJ27eHVlvW14YRngs2QxRBImC7IpYVE5I9YumXB8x2j5hmaHp5wBwBA6pWEscu/CfeVYEoq3YIYUXM5fmopcvbcQZXsgxxXrWuGQfY94iwxhYFak+Vz0IGeP9N5+tHPMeuxfkDQNuvdXjEBfOpF3LHSMByksWn8+eBQz+GN3BMWtp0QndcTORCydPMkXk5SRlU1U5PSheZkbyAYVP2u7N+84CEN7WoPUvd3PEMRjJBv36CKSD8m0i3uPora0Qby530jnKOrajuAVjp0vUjCOLFzy9P9nwrVKJ1npawojrzOO4KnF1r3JzCJW2iztjaF7pHvyLmEfHWGul9ZQyvi48gz9mriTn9dyKtZqB0W9Kmns2z2jjcQI07nOQpY24Z2bInZoUkzjV2qgZUkUp3lMerk1aktB7IYyKZ2WMlfGY95itcV/EYeCFtYhvK0pyjXFAYmiA4J0VpwYttF6FGIg1Zg3qOdl2ljp0LZ16jeiK5eRJrGOhoFYSvWKTS+K5W11EaeJeQwH1XeDzlnejvbkJY7Sz6QC0yyE47iU6itvlYXRVJcpwWwDgeEI8CHUoMcuUTz2Z+/7esUbcC7tvRZkXifHzIjnqSiSJ3415t4kuOJRbW3ccR+S5FWvWoA0IZzb8cjZ+FgoARzPPEu01EmJffZ/6ujWMeD6DE7CNlhHk0Vn7nDV6gSlnCLXou1XXmEoJx77Hyeei49UQVh7u22bjVMeJEGpw6uMQGhdkm+RmLUPc24eupVPPd03khbp6qFWeE5INjTNtMx6UXRWbSzyY45OX7NDKsbU7ils4jpJgkg9Z4oGv8B48Cp4F731uhp66bBQMzUEVayP65l1ixWOohNLmNEO4winc+cjHvsd6uNooli4+age+xlraqZwDLkgiL2Eizu5QxCPb6o31vf2/nz2FEXcMYohV9yVL2ZxnXrujuF1fnF5skCc4BG+aM95rvBaJP88O3HSuJB8GfPhQxdrc+9XXC5WhSz4HKtZjnvMU3z32PRbKE369SxVGWHrca66lpccKPcRxmepPnA3MQb1VbN6YJ65MVkxrn7LudpCn6iieC2wuBHLpiT3H+8ktiHNKJgpZzZXkwxirWJt7v/p6GX1j44WD2c2lezjmty/tu8e8x3Qhfi2c44SUCb2ldbTmWlpyrDX761QeSQhVGNQJdRSAcQpPXPWV5ISY2mMHNGH39FA87jZhdqqO4qlUmXnJlNkkNEvO8JndC6+OYg3wMR7vkOC+4aUP5V7wvYhfj1WsHasO9AA2l0NPgsf+/qV8f9d7zJYwolAjQxzfYtpOa5J1a8naa2mJcUu6W5OSuTjf5QCG7CLIJ9CBE2pSKNyIJ56eNG+IITfYtnRUYsoL5tgLHVPLqTqKMzJCAqTuG1lXkvmMsUlUx5BCJVAR/IBcziZ3X2KF3MA9MqfACIOCDb3AMOCOhPpIOiLWks14D65Y2/HM2X1K4nOMC6imOM3b6dqTjHntT6AL1SAlBTTvkjnW93mPs9mvRCnDVDtCYKCSk6P8Hwut27XX0jHDZGMUUCruAZFUTMUBqpt31PcXuWDAwQ956wz6YJOZtT3xthnoEEjf7gnd4KHwMtRyqo7iFqnNRXy89thAJGXycTOrWnMs98KKZ7Vit4TiYMgUO50L/vyYhem7mvt6gYkFWRu3vLfSbwbUht2WceNaAVFtK9aOHVd+H3fI/QvSZixmL9TiFKHALRFUY0Re1gqIm7UtPGDTkpi65Pne5z3mpGHYlNxmsGokk1Z+PptEWSww4WuvpWOGaK0gFRQi4egJH07BX+uuaX5XWGUQPbS2Ed/3oUHTgPzbF3zpjuJT46EgHsPQQhMq0AiBDAHzYakhcHDOgErtW+m6r34u4Tp6hdevEUAggmCBEAVZyLH0s2bjC0bWUbZuP9f+Fmhi1gwYZ03s1l6bib5r7S07NE+Kx8S+2+O/Vnb6DEzRxx4776dYS8eOcZXvb8GII+fRJFVizG61ZkdxnjaDy5tmtO3c+VLzqCQQhsI6CpWcFHzuSCPkkh04eHHihQpNGIFzKS5ZZUEN3FSRidAS/eLmgD3PkunEwS4BO3NiEpLxMttMhfCIoyieGMJDnGLRdNoyTkgl64P3MyaJjfdvmN9rF4b7ngXX7R2CfU55ZKGglgTfRXe8rx7pfK21tO8YNnHdFoy4YzTDt3ZHcThVMbO6alQik0fFe0hjrnJK8U8rPAljJOK64rs8DyB9SQiL+ByKSk698BhuRhGNrpOVuVYKjjNC6EkVIXKfQ5EjjDfoak2Yz+t2YlLMJQ6ZvCFjDHm1TpSbi+kSJd9DoTOfJS3vXIqKU+v/VL/nPTaH5hIvjyIriWxkV9BfY3wnh45vjbV06Fhu9HtbMOKpgDU7iouJio0SXrMXG8eLBAMvkBH2+4SxTnrIenIYbN54xj+VK4vZ8+zhm0dxnDc6w9v6cbpGkwqhwItSBaqM/1BsPpwt/hsesxOSJCu+EJuGpJEQGMMiXsuwO4ntQh4llI7mxgyPuLgxS4iKAUs6dblFAxBlTj6QKvQkhGaOs8R8KT0tvZaWGtfJ77MlI77Ww4OXZRsnuzfPqfb66EC8No38FC1rVlAZK6RNlnU7LnY5rQYYCBuyLL5TFFIu6IdabBZCZo7x+3Y+kpTmxZMxPu1EDjiF1WGD02qg/1rXwIkqNm9S0Qw0TwDqxYsuyz50PEbonqRdU00tHBl549kGDSRS/K/L6TWgFFk4hrT9PnM05p/xdnrap8ep78mXQBAQ4YAhat0MralAxPzZpWvgxjRw6Z444y1sQmqe81bh2Nuys81U3NQxXbxP+zqyRoeRG1sMZ/bD0CO8b4JcCcNeK3UPQ5u0MNou8U6oKLQObPyqPtt4PW5ryW2FYXOplXf9fv+8a2CWBi7diGeXIkoRi33QiHagGUDcdBhJ3uyhS8VL3QdZjZecwBWj9+xyWg0wnoy05LQk6ZDUoTQ4dQU/+0iWiLu2ZWaEpMJ3kcm7fe7Xr+kaWE0Dl27E6zDJWHsjsEGoEtC0qQ4oEA5i51n4k+2lelx0teU5eWNsicIkaFQxKA5JcuJAxuDs2Bf+mbhm92wbDWcFqAInnXC6dA3cqAYu3YhrVCHkQVQFZrupWunJV+5vwi/KhlvJxrPQDmhrwebgxEHQCCTEw290Jq/vx22ewloQLgrFWhH6kr/grYudg5juK5j2NCkgNbGXeymvdwqAStqFdNn39/p1XQMHa+DSjThPDfGOSj1Hay9kLZ4fwgEcSimsZFkrvsdQ8MBq/HiNYoCAcI8up9NAUjIIhUGLtFI3mh3bwMdGC04KQmh98PglskEUed4aEWiwoPlAl66BG9fApRtxCsaJAacqno1fuhahEX0hxbgZgqzCzGv8TVk1iKKXt+bH4OlJcGX7MsUmOpF0OY0G4LOV6zOwEox11yKVl4y7uRma931GmM17Xes+4u68c6REyLEumStlH/30azaigWsw4kIfjtNCHgytF5EBlvTkXasYhHJoe/WBI6LHZah5dUOdYXBrZLIM+RMj7qg91C9vI1N+UcMQ9lA9a56comD3hbgU/MiBOEFJaB7CZVMjlpzA0EEg0LKxD1HqXpRi+8OcjwauwYibDc8prq0EGJcHL06VJgMM8yt2WotefsIsPHTFHGPNot1XoQ/EQgpWP2XlXU6jAXFq1L/QRXg5wA1xWoMUQiMd6jHXLHISo9aNRtGqQrt0DWxGA9dixGuFK5WG/cUf3hNTm1mKRw/EWrb5KoE/lIelHkRdY+DvwisKf5YicDr6gfsNugbSQ+2a6BroGri1BpT1o8nVkUrITRiFIe/SNbApDVyjJ76pCeiDuVENIC5DntV2GzIoRV+KuCCcJLWHrrnRwfcf7xronnhfA9euAZDTO5Umyhk7x3SokYRGIArBJDV7ovraV8qGn7974huenD601TWgcQSGQyX0EtQafGhgq7pTMlPrQEiXLl0Dm9VAN+KbnZo+sJU1AG0CnQTVAmsuya2rD5ZL1aDgp0OkWisPq9++a2CeBroRn6evfnXXQNdA18CmNNCN+Kamow+ma6BroGtgnga6EZ+nr35110DXQNfApjTwvwf8Lqu0EynAAAAAAElFTkSuQmCC" width="184.5" height="40" /></span></div><div class = "S3"><span class = "S2">Now we want to represent the linear combination of all lines of the system</span></div><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAdIAAABQCAYAAACtb/RRAAAgAElEQVR4Xu3dB/TEWlEG8EHF3nsXe++9I/aKIHYsiGJBRREboGBBUCwodhA7KIi9YsGGXYoidlTsYsGGvZwfL8O55CXZ7G6STTZ3znnnvfff3eTeucmdOzPffHOTqFI1UDVQNVA1UDVQNXCyBm5y8i/rD6sGqgaqBqoGqgaqBqIa0voQVA1UDVQNVA1UDZyhgWpIz1Be/WnVQNVA1UDVQNVANaT1GagaqBqoGqgaqBo4QwPXYEg/MCLuGxFvEhF/cYYu6k+rBqoGqgbWqIG3iogHR8T7R8QvrnGAex/T1g3p+0TEN0fErSLix69oMZ8lIt4iIl4nIl48Iv4mIh4dEY+KiP+7onnueSovERG3iIibRYT1fmJE/HBE/N2elVLn3quBO0XE50fEe0XEI6ue1qWBLRvSd42I746I20bEw9el1rNG87wR8bhmg3Wh/2w2Vwb19yLiY+uLdJZ+1/Bjm+L9ioE8OSKeIyKeNSIeGBGfHhH/soaB1jGsSgP3iYhPiIh3jIhfWNXIdj6YrRrSmzen93tGxBdd4Ro6db57RDwgIh4TEf8TES8cEXeJiE+NCBvxV17hvPcypWeLiK+PiJ+PiIdFxFMinoag91x/TbPeb9dEIvaikzrPwxrwjHAePCdSWb9/+Cf1G0toYIuG9KUaj82JjMHZm3xaRHxhRNzmyjzxva1j33xfJCJ+vTGiwvsiElWqBlIDIlaej6dGxJtFxL9V1VxeA1szpM8UET8REa8YEa8fEX9/eRUuPgL5tCdFxDNHxKtFxD8sPoJ6w7k1IHz/1RFx14i499w3q9ffnAbsfb8cEd8WER+xudFf4YC3Zkjv1iTc5Ud/9ArXY+yUviAiPjMiPqvRx9jf1e9tQwPPHxF/2XijLxkR/7qNYddRLqiBe0SE1NaHRsS3LnjfeqsODWzJkL5pk1P6viasuecFfaOI+NWI+NMGlPS/e1bGlc79B5s8+Uc1AKQrnWad1okaAEz77Yh4voh41Yr2PlGLE/1sK4bUOH8lIl4zIl4lIv5sovlv9TIvGhF/3QyeTp7Q/PdbR8TPtiblexC/XQLABDFKvikibrdVhVzhuIV2hXiBkd6vmJ8oxOe25vudEfEBPTqQS3cN8uFNudgVqmuXU8q1/YaI+MhdamAlk96KIVWI/B0R8SUNcnUl6rvYMOSK/6OpPyw9FuCUD2rKKPyd/FhEvEvPSMHoH9F89kkR8eUXm1G9cVsDd4+Iz4uIP4+Ily4+/LCIeOOIePmIeLfm78L8SiO65F5NrtVnr9cA9aq2r0cDPxcRbxkRSBtqScyF1nULhvSmTQhDAbvNAzlBlRtYnOiEdyJfUsobNMg+f4Pw/YwehSmlyfKht+3wZqueL6eBO0TE1zUEHMJ4/90ayp2bg6U/D2EGfqgxuNC/zx0R/3W5KdU7z6CBPAwrpRKRqnIBDWzBkH58RNy/KWD/5Avo6JK3hNB99SYHwnAKb+eG+o8RAQovBHjH1iBvX+TVUCjy5rsE7ZjPsSW9QES4ZpVlNSC8/oYRobZU3hvIiHjWv7T5b6H8DMHn6ABMkJEQB6q/6hl2HrjUIztgVbk+DTw2Il43Iuph+EJru3ZDqsTjTyICclFCfU8FyO/beCQMXApw0cdFxI8UBvUrGoKG8hH6quZ7/sYQ/07P8yW36nP0dEqKqiynAQYUi9Eti1s60MhVI94QapcPJS/YUeb0WxHxGk2uvC8HXubSHxQRDlhVrk8DH9yUwkjTvPP1TW/9M1q7IUW4AKWrdlQIYy8iNyZHRpBU21R/o9k40cc9Z0S8R/P55zQw+FI3ciVv3hRtP09EdKF6UdL9c1OPii0Fb3GVZTTwcg1vMnIRrFVINnBGOzjaCIXilbzIhTKuIhPlGpZrN5QDd60sE/vEJrKzzAzrXZbUgNC/SIYDF0Q/woYqC2pg7Yb0BxqDAWz00AX1cslboQZU+kB+qmFvKusIrZm8aBpa3JslXSAgEgPJ2Craxn7SJcqJfqn5oNajLrfi1sfhCMWbMP2tI8JzXsrLNGF8niYSe95rKdY0u4AM5cAduhKEJH8mj1blOjWAWvJjIuJrG7T3dc5ypbNasyF1WhfWBZJAm7aHonTrgbD+tZtN9pUj4o87nh3eRSJscbL+dPEdodoshwFW8XJ1yUc3L53P3rMw3u3vCg+6vrB6GYZc6SO9+mHx/L+rGaXQbqKry4F7DhhQYf2fabhVy8+T+cjfhnLgD2nKYni16g0dsLrkUyLisxt0d23TtfpHqHOAEPu6Q2E6kzOH6q+ykAbWbEizXk5nF/VSexAGVAiXfEtEKHXoEnVjSQ3Wzp9lvsTvbLhOqF2SJ1if8YD6anMhQrX34iUzuFXO04DnmRdKXiki/rDjctYDDSRxYBLaL0Uzg6wbHMqBK9hHI+ke7tUnIhdCgtIBAG1VtqkBjsfLRgR8RR7WtjmTjY16zYaUQWFYGAbo0j0I48hIEsAQAJEuge608fE8ETKU8sURwcMgTql9HgYUp7rCv208/j79ekaEEn3/3/ewCDPPkYFkKAHHbHpdUob3kTEkoUJ+99capK/1eK6eHDiaQVzU1s+manPtE9EfpTG/O/Pc6+Xn1YCOQiIccCXvPe+t6tVLDazVkCpAt9EQeaJk8bn21StDtn05LSUvammVS2j0m8jO1A1wSSL3utCevvdicUMdajYB2BOQ69LPEOMmZPuTEfEOPYNJLmWGUlqj7E1qzfw/wJHDprKHLmE4E1cgn46Yocp1a0CkQ8TD8+HdrzXDC633Wg2pU5XTldAUiP9e5O0bhLL59rHQCOkJ7REe++NbysmSFi8RNF+XYMKxWZP7NqjR9vfypcy/v1ZEKLmocp4GMNFgofneiLhVx6Ugd/+o8Vq7PMmypOXHI+KdeoYDzZuf8XCF59uSUR9/P+S1njfr+uslNOCQLbcO5f02EeFZq7KABtZqSJVj2GTk8dRN7kV4KgrrGcCuzc96CdVC3PaVrGR9obIKXqt/lyKM95uNV+TvfaHzZ2/Cfbh7bxYRymja19rLukw5T0xSGKWE5yF324LOUZ0w6TpM8VCT3asLiOR3qAMxGqVYcxGItvBalJh9Y0TorJSHqynnW6+1rAYyZdPFeLbsSHZ0tzUaUmMS/pLj2SPJth6UwnBdMHb6sOn9U7MJd+W05Eey4bnDCM8nBcUiz0NNohwr4fHz/LuEIRYmenRjvHf0asw21RdqPHslLQ4oJchLvpOBBSDqK2Mo3w9IdkQl+HhThIvv1xx85GAZXaH8PrHhSg8wvmnAZ5t8vfDsGsg8qVpyHLxVFtDAGg0php0/aOa+R5Jt3MKMnTynchMhOuE+QCR0gE9tUJ9ybF0i3+k31lbpkBpFG61NW4G/shdefhLZ23R5mjbR9jVR1wG2DJXRLPCYXt0thFzlL212ymH+reGRRsogN+7wA2TUl+MqCTuAxSCqHXi8L6Ia8qMMMsPME1UGQ9Qbt8upslZ7iGbw6hbgiieUHM3y6wBkNYq0wGKv0ZAmSGLPJNvWBWoXz7AQKy8GwQL0JiYjObQhUaaCLQcQRb4ELP77m1IKG6vwcdtL0YZLO65SMh+rFpUxrTKdBiB3dTMSGVDzpzxFZEC5C7S22s8+ATjCxStC4XfeFTSQ0O3WyaGpK8rQBdzjETuoMaRVtq+BsmHFULRp+zNd0QyGDKnN2OY7JEATioCnlEQsDiESp7ifDUg+UT5yCEE5xb3OuQZ0pjA35PLSDbyTs1dOdo31hQyRLihjxMlcSByxBJAO76yP6H3M9ab6jnfQoQZZwtKkI5lvFY3IlmxTzWvO69AZYB7PngcvB2wu3g9pIYcINHkiM9ic2p1z5hzbpa/NC03ija7D8aXHN/b+l7I/xicaKAJo33P4FBlCwSg6J7rTZiJ7WvivTzyoiSoUfkpibF5Rgh1wgpbQ/LFKGvqel1rYsQ/VOMU9XKPMJR6qpZzqnlu7jtAjQAygkfDj2oSnDE3MQ0M+QWyabe9ZiFNDePWwvkt0U/GMA17tVRgixsbhFdhoCwIcB0fA27LWyCREaHQucuBUVy0Sk3sbpp9va9Iifc0btjDvY8bo0A3dvaV1bc/vEvZH9E56RU9nz5YUm6oIzoxOSziyO5uojw3t5sKYrDCjU99cgoXlFWZumwaQwSspZYjdZ665rvm6DA4PTk6NsVqz2FSzNIdhfJ2ewfqeDhk8GLL3Ho7JxbsFJhzgLJsYIB3PXdmWMPhTOtbaoQlYC31mSlfN9Zqf6XPGJkqIjOV7Chatc6536d8uZX/wVkuJiV5pSVdGW5OCsRMAO8aQ2nQSWSjXJv8yp0iSQ4veKSK0CJta5IP07zMvJ9jUwRDf7NRj2ML1UMsJkTnNf8jKB1zSIg5RK5qG5wrAKkUOeq+8pPrUagjRR1W4lmV3yMX37ICNqEUYul0/3R6rddWUIQkr2gj2tcxtjnFkr1rpGOHJLctS9oeDCJTJ9gACei9KEdXKuv0bRWHHGFLtujImPHe4tayRm+uUDLEq94dFSOcUxPAEaTc0ZJUbNIAMHXgF3WA2mF6rbkpaRDnTLxsYaEka72vI3HneexTlU1I2DpRD4KZL6kb+mGcAzW+dIM9/b+SA5AgTsawUKNnSRv58s19L7ADDgCVuy7KU/eEsOITn3ueQOVrGGFIGBlKUzG1sXr+pWXSvW0TEI0fPZNwXlQYAmwCZKBX49qbMwK9rT85n1GESB8yxDuNWa/y3tJvLMF67G077Kgjg09AKC5aN08ffcfvfBEqRVxTeFsZaqyjtQU5CbHYiJGMFEhlKXT5cznAvksxl8nw8rKVBilPqeSn7k/W3xn50KdgYQyrOngTIc4c/S6SWkEx2QplqYYT0hPaALORIgaXu3Vz8ic2pd6p7bf06jBNEpChBVx7q2PkJlTi48AgBvaaU5K91TYZxaLzlxtwVwplyXGu+Fgo5zEi8l7tMNFB5aqkZ9cdTiO5H39RcCAo3G52PvbY8PzCVVA42qTlE3TeULINVylBXHmFpoCj4AyQpU0tSrJ5kFKYezJnXm9P+lHoaGiZiC8DLXhljSLM1j4sIE5QsKmfq6EY/L93rqcE/cn6AKDhHs68mmDOSdyK0JcS1xzAfL84hJktJUMeBeysLmqosIsNsU+eqbEQOQQR6Ux6tT9TYISqwwQIUyB/ZoPcgaoQxGGVYNMPhU5Y2QcUypKI954o1MlYhXQJJmWHac6895e8dNO/RXNBhESgKaQqke5cnKHcrxwarMVeqzDsmwka2Xks6p/2hJ+hgh6BsSwgXwokoBQDJmvbKIUNqQ0WCTA5RjU3xcCYrh2sd8iyOvR8DSmlOgsmcVBKAu54QF27ZvYnwibop87cR4jgW0tWfso8+8FgdzWVIS3L9IZSiwwKPOJseK5dBt7gHSa8JHaCozM0baD9QCqarqWRKQ1quqxC0XO6a2/jZS5XayLkryVFq1SW86qzJZoCtydRSOgiiAyg+tyhL2Z9yTU7inD5kSMtuJEIkSSs316KUbcSgpAZPAUcMIptTy/uB/JfCw37J5g9zIYWPGOpFviqhz5g6vfLIdY3w/1P2p5zLkJZ0ebzN8iDk1M9D8KLgpOVl/0RE3LPwYi+i8IVv6l2CgPf+iro4SKq19c+UFHJTGlK5UGhssgX8QkltOkSpWToLImOHSG9OeVQcGtOrOhiWPOUGC/1mKfvjMKlcipzEOX3IkMorMD5ELlEh9JySdW3u4RQ9BSOJIlshXRuI+rJk/ch5lDkz+ZjbzTnBHV97LkOqy8mY8LN1R/Zh4xLmnOLZ2vFydk59KkNqXwIITICQA/b9V67ssv+raEduzO1h48tO8pC5kMRZ8+jeaub7eLnHqFSKTc3uqQIweiq96FL2x1plZOZooBHFHDKkyh+UQRAk2liN5hSegnCHfGUy0Jx7v6wbFLrsCuWVHg1QAuRwlek1MJchhcpMnth2uzDPtx6NPG3AgjwkOVgxvmXnlelnvL8rTmVIMciU5PpSDGpC1yxJbWqMQ+PVBlHYV8pMtGQOyWYTri3aVLbUO/Z+yFjOYf/qZAIaOYil7I9QPAY3h7eTOKcPGVL5MSAdot4yc4sj9XD016Y2pIpslbtgS6Korlq5MhezZ6L8oxfryB/MYUjVGCZfLvaTpLHsGxomHN4NEfoSOqoynQamMqSJKM6Rzc2mNoUGgBblJgGMAI260lLJFibUPie/t0YI0hxTGNIpdHPqNZawP9I/olVoAE/mnB4ypBZbviwX/9SibXV76lDlZ5zGhqQsR0Em39dGauzCJOn6UJ6gRH26bl/rNqUgcjX3aUoGxo4hv4fYQN6xTw8AIFPXzR47xkPfP1Sjqd4vIxjta2HEsg5ymH2F8QqiUfiNlcx9+z4wWdYb9v2+PKn7ThuFLuLCW3B4PJZAXhgS+46DW6LC2+Pw+ZprNvsahZfz4GG0Sz3yc/R9cq59HhC0Kn7kQ3KbIvqF41nno6lkrvc4aezgCtL5aI9Znj75fg+VHdlr5PZPeRbL0K7DYhuFOpUu57zOUvYH+DRZsvq4iSGzvbscyc53e8iQloth8zt1A+A228Bc71AHkbJYXsH4sZtZubBCEkK1ThzHiPBf1q6Vv0sexlOZfujBSVHYp0sP12BI1SOek0/BNKUry1jJJui+P4ZLVWmM6ESK9cjyF14E1KXifcCRY4FuadTl3JUSdck1GFLG0CZ3iowNZ+qpmweqqUkz5niPSxo7+XfRly5RHpOMOUPkEvks0peD/rHPYrmX2HcPOTCnrOXcv1nK/pQll31sevluY/hzWLyRDBlSvTAzwS8kxsidIk79TpRjaL2cVpVekHPDOdCZjNYDRgwaSUBSafXNlWGXazt0GOi7nZfNNaZEwo6Y2mq+MkdoV3cGa0eE6JW/DEk7ZNiuVcalKVTMmB4r3iXe7GNWXqZx7LyO+f5Uod0yL4cH2f5xDDuPw7NQHVEyUwLL5niPSxo7h7skeWnrDqNW7qPmmI0WunTscyWH2WnrmHXIrj5+A/PBodiaLGV/0J/q7Uv6OKfz3VZG1MnLPWRIhXAAdAiGkeQhHFoQRdOQscIWbXTsmIUsmUzOIX/gfit2lnNNesOh+z+waaTtO2PCW2PmUr/zjBqYw5BmpyB3AlB50gGllx7sIfKGun7Ha2AqQ2pfUojvoENECJJ0Y8yo8n3myWEYOvRcjLnm0HfKlNTQgS4ZjRh3Bn3K0qNyfCUhg9DwOYd3a9B3MBijN1EY63GsLGV/pNN48NKYp6YvB1G7rG8iWLWlGoPcsmBqMhUlo1TKzgNjE+vc5qSP68tVHloQuVUnPSdSRn1MWKQ8/WRfw7yPnGjWnjqNeAGOKZ1Ac8jTITwo4YM9ytSG1DMmFGvTHdNP1veeUOSvMh9S5mKsC/YfYeKxUoLV/OaQpzH2ulv83lSG1NytQfZIxSwzNmUgrG4Psd5lXfic73FZ0tKHxxAdS5o5QCDgx7aUBAQ+sy/BeRwrWHoyEgeAJ397qlwKtbuU/bGHMKBd6Ut4FgCkFPl13NQ3kj6PlDHiUfq305OY/SHjkRuSePxbNcxE8qrCbdxnucVDUqLNTq1/ytqjsV60MVFQWcgvl8ZjIQynEgonZEbxWB5RL4cDgtKbk1gzDiltI59PbUjLPBDeZOGsIfH8ocUjQrc8FTkoYDpr5LAkx2utstvRGNWifPOMeH4Aqrwrc3kaY8Zzye9MaUjtJw7EIg3Cmw7WCDWGxDNgvxEK9r5lRG3u9zhBje7TBcgTakZywpgSRg4xQ1t8D6MbD5AxPJVMIevxgTUBw9ba2advLZeyPyXQtCulR3feZ7Xn0ja9FLJ9hrTswtJ3emorIZPDwCJAIyRrOMcaNd5scvnaeCXujxHISahJ+di+cpeu6zGUQA2pj3Z4BoG+PEP75Rw7NjRgPJ2TWDOacdn4vVhC3kBYNi1sU1vppTm1IZXXyPZuvA0dL7qEkePR3L3RI93JaQk5lZLk2GNCxO37eOGAcJyij+3/iDDEuyPqw3tgNFxH67CtbYBTGlI69g57xm1gol2aKnfVk8IfMB53bA5GWJw8H+286lzvcfkstjdkG7HOIsgXrDPRyjGxIF3PrEOZd92+dArgMgFVS/SPHrsHHvO9pexP2VKxs2F3M2hNMdiHXh7v0pCKhTN8REg0kYcuUCId+xTiJOVGJfFBxrnHJrydyBgG/z6GzcRYeb88Wv+G+BOK9tIJp/YJ7zLLNTzcCVDwIDtACMWAoSeyywtTNoUe+3DwcGzep7BmWCNzYNyJzUGoBhjL5q2kRmhp7ZvuFIaUF0qPBKw/ydFttu1GzzwaJ04bEmNKREsA2rq6CnnGeaantFXLspohariuZ8VG+bjGk/W5OmZeMoPqMIgBZ+0lUeW8pjakru2A4QCr4QBRW+hQa3OzrsoXfCayIHQvIqUUqkvmeo8d4I3L80PgLKyrd1RkjdEU2XPgI95nRs4aM7KleN/tubzvvjKanuk9/c9Jr+gwJjq4BbmE/SnTB33dxoxLjl26ILug3UifpSHFa3lMr7++xSmN5q81L4IH3iYxRrDNOGEyWImmOvQ7k+yCJQMBOJ31Sclq1PWdNJzZKePU3pzm5HBwEmtGYxDMA+BLGNNhgxcE3OUEbOOwfmsOKU5hSIXEskvD0DPhUCE3Lv9hs/Icekb6WiHxGuTGhd9OKfPKnBQjfSwdmudW/au5QfxaQ4w3wsyMgsPtMSVBh96VOT+fw5Aar30KgMZGJsSGiEP4FkDEAcihF7gQFmMI3TvnewzRbz+BDHdAgv4W9eAdK7FS9qL8pRTUdEkXmH+HHBVV8/2+muxDa8iA8n6HSnEOXWPpzy9hf5JedAj7kohsUcXs8jNoSKdWHMPBY/JQOFWOFWUrvI2heryx15rqe4yXk+UpJTmKeYXrTmbNODAJHpcX1qk3WXummvderkOHQATKvU7RYebIpmxJRvdC0g5QCAoevpfFmHGea36Pc9rJ2StUnTznx6okG3Ew7CJWe5Qx9ifpRR20kV90CVwLz1UYOFvTLWpIndCABbSuuu0RK5l1PUIlrrEGEUqVuM9a02PGlDVdfawZx1yr77tOnriQhXGcRqscpwFhfcYQH+8pUH2ernwezxYTz1QifyqsZFMQ5uNhVzldA2t/j80sOXvtGwz/sQL8JhqTXjxPfY9yyP6UQKMhlqmHNpUWgxS5h7h2z1mADOUdA113P0lfOZFTSk3OGW/fb+Wr5CvGUNB1XSMRdH2sGVOMOY312DKjKe55TdfI7g+MYXKUjp1fUmgiWVcqMLXkxnpsWc7U49j69bbwHtOxyJWyC5EsZV3HSln5AEUuF7tHOWR/Sha2IQyPdIVUo3B9Lw5lTkMKwi1Hicj5GP5UwIGkbTu1lnTKByeb5A61kYOodQrsemgzN9LHmjHFWG3mAAo8IkAx4fQq4zUAhCS00+dRAqHZoHie7Tw0T1H0BL4AmGVqyY0RP7GN8RiGn6nHsuXrbeE9pl+5VdGvJKM4VueZrwfGkoraqwzZH9UhbIzD1dB7Kw+vDBS9ayKuO/U5pyHlweEotNEA24yFcashEsIyiVNDbVM+PJmn8oBC2wGllCLcK/wmpAdBWLZ/8j2wfQt2MmvGyMlALDLoSgCgeAlgRsmzycsXougjjRcWcujxHQ9O1tKOHMImv5YepagDY2od2yVFkJWeRXql31IAQrR7OpWD+ZDSIEKzoN7zZZ275F6tfsGQzDybLskx52cPaYBrh8ay5c+38B4DI8rb9XK6jlgAwDTPKKwJzuK9Spf9ga532PUPhLW9Doitj7RHhIoRxb0OeMqodgI65zSkmfC2kH0FyH2LnC2JbGDZcPVSD4QcL3Qs4enx+EpRZiFZjcbMBlWCQqCVGV5AllPQoMfMOUFaZU7ag/ChTa5ZofjQWjAoDgpAVcYKQboHcWgoeaChBxnGUqBxIT4dLJKlKj8HCIGuPRXRfUjH1oVhly8dOlg66IngKD/jQXvulPJ0haPUM/vH82pDmesQcGhuS36+hffYmkCSDiJEDygNF7gDobrqLLdZUs9ruVeX/ZEm8Q7xRlVAOEAOlQ36blme1O53/PS5zmlIz1FostAAK82RdzpnbF2/pUe5DYovvb0kST/UMmmK8WQT3C6qK+UUjEDSG4oStGuDeVu3b0oxGOUqz6gBhtLpNclG8lMtqjBjyWkJr88hiS4cs8GWHWYY/XZ9bTk+aHLjdnAqmb3mmMMWrnnp9xjCFje4unhVC8eKPJ6QLnDaGtJix45/s99fqyFVLqOgmZxDXr/EwhgfQJEHnzcAFJIeTtatTVUWYb0w73hJhMqFbZUYkaylFfoTAmxLWXzcRlJj/bFJQ1e3vbEldLj2e/Dq5e6Fi4RL79wMmDcnJAzkxZs4R3icQvN6VjKcPIuk5eRd2iS7Qsvte5acsqI57YL//L5DlVCVEzne4lOaTJwz37X9dsn32Nzl43mfDtspGdmSCjqFsSxrHj0/vKcqC2lgrYbUuHh2HgYhKwxJaxW5XKwl2WmdQUImgYVHqPRbJwpPO1wIRZQlQWL7Otxoz5SeEdYeLB1tEebjldo0AVZcj8fP+NMvFBvPucqNNcDoJIsUffHgrLnaXZ4qDlWAo1MFohuRQ8mq5PlXliPSkQZVcX+yj/XdC2lBtpP75gYF3/XdjJagIzyWP/rUea75d0u9x6kD+hc9kHKxBgjqhR67Ugtj9ZZt2h5UdLMa+9v6vTM0sFZDakqZe9pKxxS5UhsdjwWwSDd1m6N/zmUcwnwj94rNCPCEpyH8CpzgxfO5Qwc02lAbuAwd0a/NlgFVZ2bcY5oKnPGoXcVPnfjp0GEGqw4mJP9/Tpuqkl1LhEG/Soch9xDpsMEnLaKwn4PTkGSZh+905fTzt5k+QfgAXV/lBg3M+R6XOpZeUTMvWiUPLgIHMHZK7WheFyweonwAAAXiSURBVNBRxArIE86kykIaWLMhzfY9vC6UYBnCXEg1q7kN4yhcy5OEfuYBtctbAITyBWRwMeF0ifAgrzQ5QSFUGVLGeO1cvatZkAkH4gCUuTARBXSBJbrd+ynCIfRO5GfH0AVaY6UypK8eMVGNWMTcu8q2NQBgJB2gggDZQC2TWnA912xIqSF70g0x8y+orovcCm2dsA+Rw5TfbEuZUz7ktZRNiCHbNBoYy4N8EQVc6U29e7wQgCChW+jhdumUqaMsxKdMulp0dalHCkBBOulqC8fLRY4vD+eQeko+7kqXZbPTyo4v+GAdvqosqIG1G1KEzgAWe2XskXdVoJ2oW6U3XX1hs6uFRweydIgWLFmQ8jGrtIILvnDFrRjQ7EIDiq/VYJdkByWfiSSMoQnM9oV+09ViLttHufaYJgCX0VC961gN3LTBlNgnRCJErqosqIG1G1IEAcJUTs1yCmW934JqutiteItZpgK0BD3aJfdtwELCs0LBjG+XyLupaeWB0inRpku4uMqyGkiQl7sqOwIQ6RKUhdiN+tDYXb+Rd8u+nV3lUBoVK7EAiJNTr7JtDegqg0ENCjjz6due0cZGv3ZDSp2K3QGP1kDOsPTyliHbIa5V1HWQo4ykmsYuAUYCZhHS45UqSk4aMjlWXn+V5TRQhmytmbVri5y2Wk8gM+VLnoExgh0MGMrvYAzk1zOS4TkQQgaGc7CqufExGl33dxyGoIDtAV2Nz9c9+isY3RYMqdIDHLbyOrrMP/kK9D52CjxyICshXmjOzJWVvy+ZeeRSdTFpi40UwpQuUf/xWJOT03cZ2EEuybEDrt8brQEgnyS+6CueL9foELlC+8Z5uPL3kow/W75JB0zRf3j0hOsXZ9FAljFp8SVkX+UCGtiCIaWWzPkgOOCh7kmS8kv4Fl9oW7IAXxE2Yvx2Gy9eCSg8FDRjmYhfBABqH/2GQJBCclZZRgNqRh1oeI9duvduOuAI056ySWY7QrPJQ5hnAPXjExtv9NyyrGU0Ve8ypIGsH7e255RhVS2foYGtGFIeGdYPzC/CUTaCvYiwnzyXdj7tUBx92BgZS7yp8iSlqE/zN6w78qCMcik4hBMFXAvzl3+i7trUDmrjBlhXSrYTFKLlUR67SWaDaNfUr9bzISqhiXk9NC2/1nPc0TstJdP1/Mxxv3rNHg1sxZAavk4mwlVO5311kte60EpWsJ4oxlfAL6/FwDKSSBl8Lo/cFqFg7DhKIDDktIWhfWxBxI4JRw9Eza0Z7irzagDaEuGIFl+3jAgdWxwaAZGg1eU3b31i/lr+W00hwZIkooF4AxGHa1bZtgZEMpRPiWwI++8p5bW6lduSIaW8bBG0R+YOZSoAJ/LEBIOJHBsD24W8zLZRh2pwbeDtcpnBbvCre4q3PSDvINSu/LacuB6S8AAPawjMz2llV3bAEPKH1hbVkAaosm0N3K05VNsLHcCqXFADWzOkQpiPako3nMLm6rZxwSU5eGseCxo4J9BKpHBQXZv6gvZn6CXRQE7BTIPlqvQ++wg9NqWkOtincWlD5wKO4ciucmENbM2QUhf6K6UbvDDkA1WqBqoGujVgkxXSJUME9lV/29GArjHwItD8yl3qYXoFa7dFQ0ptio61DVNXJ7RZpWpgjxoQobldRDyiA4Dn3eaxADHZeJVJtBHde9TZ1ucMOAZ9DyPRRSm59fltcvxbNaSUfYemjRVWD4CNKlUDe9MANO8vN8YScjMFlSSOZsQbQGPaviF2qLJtDQAaflRzKNoby9uqV27LhpRiATSgVdGdVWaeVT9qdXAzaOBjmsMkrECWMcmfKXHxbit3kSOFxK6ybQ1gwkJIj9bxN7c9lesb/dYNqRVB9i28qzlxH8fs9a1cnVHVwA21oUqfcDIDKmk27h14fNO/9sET9MKter68BkQUNDbAXCRMX2VlGrgGQ7oyldbhVA1UDVQNVA3sSQPVkO5ptetcqwaqBqoGqgYm10A1pJOrtF6waqBqoGqgamBPGvh/9q79uiTllvgAAAAASUVORK5CYII=" width="233" height="40" /></span></div><div class = "S3"><span class = "S2">with linear combination of total differentials</span></div><div class = "S4"><span style="vertical-align:-18px"><img src="data:image/png;base64,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" width="251" height="43" /></span></div><div class = "S3"><span class = "S2"> Matching all partial derivatives we obtain</span></div><div class = "S4"><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="104.5" height="21" /></span></div><div class = "S4"><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="106.5" height="21" /></span></div><div class = "S3"><span class = "S2">Multiplying the first set of equations by </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="21" height="18" /></span><span class = "S2"> and the second set by </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="23" height="18" /></span><span class = "S2"> and subtracting them leads to</span></div><div class = "S4"><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="150" height="21" /></span></div><div class = "S3"><span class = "S2">and from the solvability condition we obtain </span></div><div class = "S4"><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="166" height="21" /></span></div><div class = "S3"><span class = "S2">which can be solved for </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="23" height="36" /></span><span class = "S2"> or </span><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="23" height="36" /></span><span class = "S2"> . Integration then leads to characteristics </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="62.5" height="20" /></span><span class = "S2"> or </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="60.5" height="20" /></span><span class = "S2">.</span></div><div class = "S3"><span class = "S2"></span></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><h1 class = "S1"><span class = "S2">1.5 Examples</span></h1><h2 class = "S5"><span class = "S2">Second order PDE: Time-dependent diffusion</span></h2><div class = "S4"><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="83" height="38" /></span></div><div class = "S3"><span class = "S2">Considering one spatial variable </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="17" height="20" /></span><span class = "S2"> and taking time as the second independent variable </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="17" height="20" /></span><span class = "S2">, we can rewrite the equation as</span></div><div class = "S4"><span style="vertical-align:-18px"><img src="data:image/png;base64,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" width="217.5" height="47" /></span></div><div class = "S3"><span class = "S2">with</span></div><div class = 'LineNodeBlock contiguous'><div class = 'inlineWrapper'><div class = "S11 lineNode"><span class = "S12">a = [ 1 0</span></div></div><div class = 'inlineWrapper'><div class = "S11 lineNode"><span class = "S12">      0 0 ];</span></div></div><div class = 'inlineWrapper'><div class = "S11 lineNode"><span class = "S12">b = [ 0</span></div></div><div class = 'inlineWrapper'><div class = "S11 lineNode"><span class = "S12">      1 ];</span></div></div><div class = 'inlineWrapper outputs'><div class = "S11 lineNode"><span class = "S12">eigs(a)</span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 14px;"><div class="inlineElement embeddedOutputsMatrixElement" uid="861C688F" data-testid="output_0" data-width="1141" style="width: 1171px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">ans = </span><span class="veVariableValueSummary" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:43.5,&quot;totalColumns&quot;:&quot;1&quot;,&quot;totalRows&quot;:&quot;2&quot;,&quot;charsPerColumn&quot;:6}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 45.5px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">     0
     1
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class = 'inlineWrapper outputs'><div class = "S11 lineNode"><span class = "S12">rank([a b])</span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 14px;"><div class="inlineElement embeddedOutputsVariableElement" uid="B211A64D" data-testid="output_1" style="width: 1171px; white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">ans = 2</div></div></div></div></div><div class = "S14"><span class = "S2">As one eigenvalue is zero and the rank of the composite matrix </span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="26.5" height="19" /></span><span class = "S2"> equals the number of independent variables, we conclude that the equation is parabolic.</span></div><div class = "S3"><span class = "S8">Task: </span><span class = "S2">Repeat this example for higher number of spatial dimensions.</span></div><div class = "S3"><span class = "S2"></span></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><h1 class = "S1"><span class = "S2">Recommended literature</span></h1><div class = "S3"><a href = "https://tiss.tuwien.ac.at/course/courseDetails.xhtml?courseNr=302017"><span class = "S0">Lecture notes for the course LVA-Nr. 302.017: Grundlagen der numerischen Methoden der Stömungs- und Wärmetechnik, TU Wien</span></a></div><div class = "S3"><a href = "http://how.gi.alaska.edu/ao/sim/chapters/chap3.pdf"><span class = "S0">Otto, A. (2011), 'Classification of PDE's and Related Properties' In </span><span class = "S16">Methods of Numerical Simulation in Fluids and Plasmas</span><span class = "S0">. Lecture notes, University of Alaska</span></a></div><div class = "S3"><span class = "S2">Fletcher, C. (1998), 'Partial Differential Equations' In </span><span class = "S8">Computational Techniques for Fluid Dynamics 1: Fundamental and General Techniques, </span><span class = "S2">Springer, Berlin.</span></div><div class = "S3"><a href = "https://www.springer.com/de/book/9783540678533"><span class = "S0">Wesseling, P. (2001) 'Classification of partial differential equations' In Principles of Computational Fluid Dynamics, Vol. 29 of Springer Series in Computational Mathematics, Springer, Berlin.</span></a></div><div class = "S3"><span class = "S2"></span></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><h1 class = "S1"><span class = "S2">Appendix A: Characteristics of first order PDE with multiple independent variables</span></h1><div class = "S3"><span class = "S2">Consider a generic first-order PDE</span></div><div class = "S4"><span style="vertical-align:-18px"><img src="data:image/png;base64,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" width="124.5" height="47" /></span></div><div class = "S3"><span class = "S2">which we want to replace by an ODE, e.g.</span></div><div class = "S4"><span style="vertical-align:-18px"><img src="data:image/png;base64,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" width="89" height="38" /></span></div><div class = "S3"><span class = "S2">where the total derivative is defined as</span></div><div class = "S4"><span style="vertical-align:-18px"><img src="data:image/png;base64,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" width="225" height="41" /></span></div><div class = "S3"><span class = "S2">Rearranging the original PDE to the form</span></div><div class = "S4"><span style="vertical-align:-18px"><img src="data:image/png;base64,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" width="166.5" height="47" /></span></div><div class = "S3"><span class = "S2">we obtain </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="109" height="21" /></span><span class = "S2">. Note that in case of </span><span style="vertical-align:-7px"><img src="data:image/png;base64,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" width="140" height="21" /></span><span class = "S2"> the characteristic is a curve evolving in time.</span></div></div></div>
<!-- 
##### SOURCE BEGIN #####
%% Exercise 1: Classification of partial differential equations
% Date of exercise: 8th May 2019
% 
% Last update: 16th May 2019 by Lukas Babor
%% 1.1 Classification of PDEs
% Consider an example of a partial differential equation
% 
% $$\sum\limits_{i, j=1}^{d}\frac{\partial}{\partial x_i} \left( a_{i, j} 
% \frac{\partial u}{\partial x_j} \right) + \sum\limits_{i=1}^d b_i \frac{\partial 
% u}{\partial x_i} + c u = f$$
% 
% The aim of this chapter is to introduce a classification which would describe 
% the properties of the equation and its solution. 
%% Basic classification
% * Order
% * Number of independent variables
% * Linearity
% * Constant/variable coefficients
% * Homogenity
% 
% With this criteria we can describe our example as: _Linear, non-homogeneous 
% _( $f \neq 0$ ) _, second-order PDE with _$d$ _independent variables and constant 
% coefficients _( $a_{i,j}, b_i, c = const.$ ).
% 
% In addition, second order PDEs and some systems of PDEs can be divided 
% into three types: elliptic, parabolic and hyperbolic. The type of equation determines 
% certain properties of the solution and it imposes restrictions on boundary conditions 
% and discretization methods which can be used to solve it numerically. The next 
% section illustrates each type on examples. It is followed by some methods to 
% determine the type of an equation. 
% 
% 
%% 1.2 Properties of different types of PDEs
%% Elliptic
% * Boundary conditions must be prescribed all along the boundary
% * Local change of the boundary data influences the solution everywhere
% * No propagation of waves 
% * No real characteristics
%% Parabolic
% * Parabolic equations arise by adding time dependence $\frac{\partial u}{\partial 
% t}$ to equation which was originally elliptic.
% * Waves can propagate in one specific direction only, corresponding to time-like 
% variable
% * Equivalent system of $n$ first-order equations has 1 to $n-1$ real characteristics
% * Boundary conditions are required in space, initial condition in time
% * Local change in boundary conditions affects immediately the solution at 
% all positions in space, but only in future times
%% Hyperbolic
% * Waves can reflect from boundaries
% * Equivalent system of $n$ first-order equations has $n$ real characteristics.
% * Boundary conditions are required in space, initial condition in time
% * Information about local change of boundary condition travels through space 
% with finite speed.
%% Types of boundary conditions
% * Dirichlet:  $u = u_D \quad \text{on } \Gamma_D$ 
% * Newton:  $\frac{\partial u}{\partial n} = g \quad \text{on } \Gamma_N$ 
% * Robin: $\frac{\partial u}{\partial n}  + \alpha u = g_R \quad \text{on } 
% \Gamma_R$ 
% * periodic: $u(\Gamma_{P1}) = u(\Gamma_{P2})$
% 
% 
%% 1.3 Methods to determine the type of PDE
%% Second order PDE
% $$\sum\limits_{i, j=1}^{d}\frac{\partial}{\partial x_i} \left( a_{i, j} \frac{\partial 
% u}{\partial x_j} \right) + \sum\limits_{i=1}^d b_i \frac{\partial u}{\partial 
% x_i} + c u = f$$
% 
% Second order PDEs describe a wide range of physical phenomena including 
% fluid dynamics and heat transfer. It is convenient to classify them in terms 
% of the coefficients multiplying the derivatives. Replacing $\frac{\partial}{\partial 
% x_i}$ by $s_i$ we can write the characteristic equation of the left hand side 
% as
% 
% $$\sum\limits_{i,j=1}^d s_i a_{i,j} s_j + \sum\limits_{i=1}^d b_i s_i + 
% c = 0$$
% 
% The PDE is:
% 
% # *elliptic* if the matrix $a_{i,j}$ is (positive or negative) definite, i.e. 
% its eigenvalues are non-zero and all have the same sign.
% # *hyperbolic* if the matrix $a_{i,j}$ has non-zero eigenvalues and all but 
% one have the same sign.
% # *parabolic *if the matrix $a_{i,j}$ is semi-definite (i.e. exactly one eigenvalue 
% is zero, while the others have the same sign) and $rank(a_{i,j},b_i)=d$
%% 
%% Second order PDE with two independent variables
% Note that in case of two independent variables we can write the second-order 
% terms as
% 
% $$\sum\limits_{i, j=1}^{d}\frac{\partial}{\partial x_i} \left( a_{i, j} 
% \frac{\partial u}{\partial x_j} \right) \equiv A \frac{\partial^2 u}{\partial 
% x^2_1} + B \frac{\partial^2 u}{\partial x_1 \partial x_2} + C \frac{\partial^2 
% u}{\partial x^2_2} $$
% 
% The equation can then be classified as 
% 
% * *elliptic *for $B^2 - 4AC < 0$
% * *hyperbolic *for $B^2 - 4AC > 0$
% * *parabolic *for $B^2 - 4AC = 0$
% 
% _Remark: _In case of variable coefficients, the equation can change its 
% type in different parts of the domain.
% 
% _Remark: _In more than two independent variables, it is often useful to 
% determine how the solution behaves on a certain two-dimensional sub-space. One 
% can "freeze" $d-2$ independent variables on some fixed values (i.e. remove derivatives 
% with respect to the "frozen" variables) and determine the type of the resulting 
% reduced equation with two independent variables. 
%% 
%% Systems of first-order PDEs with two independent variables
% The general form can be written in Einstein's summation notation, 
% 
% $$A_{i,j} \frac{\partial u_j}{\partial x} + B_{i,j} \frac{\partial u_j}{\partial 
% y} + C_{i,j} u_j = f_i$$
% 
% or in matrix-vector form, 
% 
% $$\mathbf{A} \partial_x \mathbf{u} + \mathbf{B} \partial_y \mathbf{u} + 
% \mathbf{C u}  = \mathbf{f}$$
% 
% where $\mathbf{u}$ is the vector of unknowns. The characteristics can be 
% computed from the matrices $\mathbf{A}$ and $\mathbf{B}$ by solving
% 
% $$\det \left( \mathbf{A}^t \d y - \mathbf{B}^t \d x \right) = 0$$
% 
% for either $\frac{\d y}{\d x}$ or $\frac{\d x}{\d y}$. We can conclude 
% that the PDE is
% 
% * *elliptic *for no real solutions
% * *hyperbolic *for $n$ real solutions
% * *parabolic *for 1 to $n-1$ real and no complex solutions
% 
% where $n$ is the number of equations. Note that there are cases at which 
% this method cannot determine the type of the system.
%% 
%% Systems of PDEs with three (or more) independent variables
% $$A_{i,j} \frac{\partial u_j}{\partial x} + B_{i,j} \frac{\partial u_j}{\partial 
% y} + C_{i,j} \frac{\partial u_j}{\partial z} = 0$$
% 
% The characteristics are always one dimension lower than the number of independent 
% variables. That means that with three independent variables the characteristics 
% are surfaces. The solution of the equation
% 
% $$\det \left( \mathbf{A}^t \lambda_x + \mathbf{B}^t \lambda_y + \mathbf{C}^t 
% \lambda_z \right) = 0$$
% 
% now provides the normal vectors $\mathbf{\lambda} = (\lambda_x, \lambda_y, 
% \lambda_z)$. One can divide the above equation by $\lambda_x$, $\lambda_y$ or 
% $\lambda_z$, provided it does not destroy any solutions. The classification 
% is then the same as in the case of two independent variables.
%% 
%% Higher order PDEs
% It is always possible to transform a higer order equation to a system of first-order 
% equations by introducing auxiliary dependent variables. The methods above can 
% then be used to investigate the type of the equation. 
% 
% _Attention: _Transformation to the system of first-order equation can create 
% _spurious solutions -_ the solution of the original higher-order equation solves 
% the first-order system, but not all solutions of the first-order system solve 
% the original higher-order equation. Therefore, the type of the system can differ 
% from the original equation. Second order PDEs should therefore be prefferentially 
% analysed in terms of the coefficient matrix.
% 
% 
%% 1.4 Method of characteristics
% Characteristics of an equation with $d$ independent variables are $d-1$ dimensional 
% objects (curves for $d=2$, surfaces for $d=3$), such that the propagation of 
% the solution along these objects can be described by ODE (i.e. partial derivatives 
% can be replaced by total differentials). The number of real characteristics 
% determines the type of an equation. 
%% *First order equations* 
% Consider a first-order PDE with two independent variables
% 
% $$a \frac{\partial u}{\partial t} + b \frac{\partial u}{\partial x} = f$$
% 
% which we want to replace by an ODE
% 
% $$\frac{\d u}{\d t} = g$$
% 
% where the total derivative is defined as
% 
% $$\frac{\d u}{\d t} = \frac{\partial u}{\partial t} + \frac{\d x}{\d t} 
% \frac{\partial u}{\partial x}$$
% 
% on the characteristic curve $x=x(t)$. Dividing the original PDE by $a$ 
% we obtain the same form as the definition of the total derivative
% 
% $$\frac{f}{a} = \frac{\partial u}{\partial t} + \frac{b}{a} \frac{\partial 
% u}{\partial x}$$
% 
% thus obtaining $\d u / \d t = f / a$ and $\d x / \d t = b / a$. After integration 
% we obtain the characteristic
% 
% $$x(t) = x_0 + \frac{b}{a} t$$
% 
% along which the solution evolves as  
% 
% $$u = u_0 + \frac{f}{a} t$$
% 
% The extension of this example to the case of multiple independent variables 
% should be evident (see Appendix A). Finding the characteristics of higher order 
% equations or systems of equations follows similar arguments.
% 
% 
%% *Second order equations*
% Let us now consider the case of second-order PDE with two independent variables. 
% 
% $$A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial 
% x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial 
% x} + E \frac{\partial u}{\partial y} + F u = g$$
% 
% First we introduce auxiliary unknowns $P=\frac{\partial u}{\partial x}$ 
% , $Q = \frac{\partial u}{\partial y}$ with total differentials
% 
% $$\d P = \frac{\partial^2 u}{\partial x^2} \d x + \frac{\partial^2 u}{\partial 
% x \partial y} \d y$$
% 
% $$\d Q = \frac{\partial^2 u}{\partial x \partial y} \d x + \frac{\partial^2 
% u}{\partial y^2} \d y$$
% 
% Note that this corresponds to converting the second-order equation into 
% a system of first-order equations. Next we want to rewrite the original equation 
% in terms of total derivatives $\frac{\d P}{\d x} , \frac{\d Q}{\d y}$ . We try 
% to expand
% 
% $$A \frac{\d P}{\d x} + C \frac{\d Q}{\d y} = A \frac{\partial^2 u}{\partial 
% x^2} + \left( A \frac{\d y}{\d x} + C \frac{\d x}{\d y} \right) \frac{\partial^2 
% u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2}$$
% 
% and compare it to the original equation. We obtain 
% 
% $$A \frac{\d y}{\d x} + C \frac{\d x}{\d y} = B$$
% 
% which after multiplication by $ \frac{\d y}{\d x} $ or $ \frac{\d x}{\d 
% y} $ , leads to
% 
% $$A \left( \frac{\d y}{\d x} \right)^2 - B \frac{\d y}{\d x} + C = 0$$
% 
% thus giving us the criteria 
% 
% * $B^2 - 4 A C <0 \rightarrow$ no real characteristics $\rightarrow$ elliptic
% * $B^2 - 4 A C = 0 \rightarrow$ one real characteristic $\rightarrow$ parabolic
% * $B^2 - 4 A C > 0 \rightarrow$ two real characteristics $\rightarrow$ hyperbolic
%% Systems of first order equations with two independent variables
% $$A_{i,j} \frac{\partial u_j}{\partial x} + B_{i,j} \frac{\partial u_j}{\partial 
% y} + C_{i,j} u_j = f_i$$
% 
% Now we want to represent the linear combination of all lines of the system
% 
% $$L_i \left( A_{i,j} \frac{\partial u_j}{\partial x} + B_{i,j} \frac{\partial 
% u_j}{\partial y} + C_{i,j} u_j \right) = L_i f_i$$
% 
% with linear combination of total differentials
% 
% $$\sum\limits_i m_i \d u_i = \sum\limits_i m_i \left( \frac{\partial u_i}{\partial 
% x} \d x + \frac{\partial u_i}{\partial y} \d y \right)$$
% 
%  Matching all partial derivatives we obtain
% 
% $$L_i\ A_{i,j} = m_j\ \d x$$
% 
% $$L_i\ B_{i,j} = m_j\ \d y$$
% 
% Multiplying the first set of equations by $\d x$ and the second set by 
% $\d y$ and subtracting them leads to
% 
% $$\left( \mathbf{A}^T \d y - \mathbf{B}^T \d x \right) \mathbf{L} = \mathbf{0}$$
% 
% and from the solvability condition we obtain 
% 
% $$\det \left( \mathbf{A}^T \d y - \mathbf{B}^T \d x \right) = 0$$
% 
% which can be solved for $\frac{\d y}{\d x}$ or $\frac{\d x}{\d y}$ . Integration 
% then leads to characteristics $y=y(x)$ or $x = x(y)$.
% 
% 
%% 1.5 Examples
%% Second order PDE: Time-dependent diffusion
% $$\frac{\partial u}{\partial t} - \Delta u = f$$
% 
% Considering one spatial variable $x_1$ and taking time as the second independent 
% variable $x_2$, we can rewrite the equation as
% 
% $$\sum\limits_{i,j=1}^{2}\frac{\partial}{\partial x_i} \left( a_{i,j} \frac{\partial 
% u}{\partial x_j} \right) + \sum\limits_{i=1}^{d} b_i \frac{\partial u}{\partial 
% x_i} = f$$
% 
% with
%%
a = [ 1 0
      0 0 ];
b = [ 0
      1 ];
eigs(a)
rank([a b])
%% 
% As one eigenvalue is zero and the rank of the composite matrix $a|b$ equals 
% the number of independent variables, we conclude that the equation is parabolic.
% 
% _Task: _Repeat this example for higher number of spatial dimensions.
% 
% 
%% Recommended literature
% <https://tiss.tuwien.ac.at/course/courseDetails.xhtml?courseNr=302017 Lecture 
% notes for the course LVA-Nr. 302.017: Grundlagen der numerischen Methoden der 
% Stömungs- und Wärmetechnik, TU Wien>
% 
% <http://how.gi.alaska.edu/ao/sim/chapters/chap3.pdf Otto, A. (2011), 'Classification 
% of PDE's and Related Properties' In >
% 
% Fletcher, C. (1998), 'Partial Differential Equations' In _Computational 
% Techniques for Fluid Dynamics 1: Fundamental and General Techniques, _Springer, 
% Berlin.
% 
% <https://www.springer.com/de/book/9783540678533 Wesseling, P. (2001) 'Classification 
% of partial differential equations' In Principles of Computational Fluid Dynamics, 
% Vol. 29 of Springer Series in Computational Mathematics, Springer, Berlin.>
% 
% 
%% Appendix A: Characteristics of first order PDE with multiple independent variables
% Consider a generic first-order PDE
% 
% $$\sum\limits_{i=1}^d b_i \frac{\partial u}{\partial x_i} + c u = f$$
% 
% which we want to replace by an ODE, e.g.
% 
% $$\frac{\d u}{\d x_1} + e u = g$$
% 
% where the total derivative is defined as
% 
% $$\frac{\d u}{\d x_1} = \frac{\partial u}{\partial x_1} + \frac{\d x_2}{\d 
% x_1} \frac{\partial u}{\partial x_2} + \frac{\d x_3}{\d x_1} \frac{\partial 
% u}{\partial x_3} + \dots $$
% 
% Rearranging the original PDE to the form
% 
% $$\frac{\partial u}{\partial x_1} + \sum\limits_{i=2}^{d} \frac{b_i}{b_1} 
% \frac{\partial u}{\partial x_i} + \frac{c}{b_1} = \frac{f}{b_1}$$
% 
% we obtain $\d x_i / \d x_1 = b_i / b_1$. Note that in case of $x_1 = t, 
% x_2 = x, x_3 = y$ the characteristic is a curve evolving in time.
##### SOURCE END #####
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