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Differential Equations Lecture Work Solutions 87

Differential Equations Lecture Work Solutions 87 - 18 2 u =...

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18. 2 u =0 , 0 x π, 0 y π u ( x, 0) = sin x +2sin2 x u ( x, π )=0 u (0 ,y )= u ( π, y )=0 Separation of variables leads to X 0 + λX =0 and Y 0 λY =0 The last boundary condition dictates X (0) = X ( π )=0 and we can solve the ODE for X λ n = n 2 ,n =1 , 2 ,... X n =s in nx, n =1 , 2 ,... Thus the ODE for Y becomes Y 0 n n 2 Y n =0 with a bounday condition coming from next to last condition Y n ( π )=0 Ths solution is Y n ( y )=s inh n ( y π ) Thus we have u ( x, y )= X n =1 a n sinh n ( y
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