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Differential Equations Lecture Work Solutions 80

Differential Equations Lecture Work Solutions 80 - 13 ut =...

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13. u t = 2 u xx u (0 , t ) = 0 u x ( L, t ) = 0 u ( x, 0) = sin 3 2 π L x u = XT X ˙ T = 2 X T ˙ T 2 T = X X = λ X + λ X = 0 ˙ T + 2 λ T = 0 X (0) = 0 X ( L ) = 0 X n = sin n + 1 2 π L x n = 0 , 1 , · · · λ n = n + 1 2 π L 2 u ( x, t ) = n =1 a n e 2 [( ( n + 1 2 ) π L ] 2 t sin n + 1 2 π L x At t = 0 u ( x, 0) = n =1 a n sin n + 1 2 π L x But also u ( x,
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