Differential Equations Lecture Work Solutions 154

Differential Equations Lecture Work Solutions 154 - b utt...

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b. u tt c 2 u xx = xt u ( x, 0) = sin x u t ( x, 0) = 0 u x (0 ,t )= t u x ( L, t t 2 w ( x, t t 2 t 2 L x 2 + tx w t = 2 t 1 2 L x 2 + x w tt = x 2 L w xx = t 2 t L Thus w ( x, 0) = 0 ,w t ( x, 0) = x x 2 2 L Let v = u w then v tt c 2 v xx = xt x 2 L + c 2 t 2 t L | {z } s ( x,t ) v x (0 v x ( L, t )=0 λ n = L 2 φ n =co s L xn =0 , 1 , 2 , ··· v ( x, 0) = sin x since w ( x, 0) = 0 v t ( x, 0) = 0 x + x 2 2 L v ( x, t 1 2 v 0 ( t )+ X n =1 v n ( t )cos L x s ( x, t 1 2 s 0 ( t X n =1 s n ( t L x s n ( t 1 L Z L 0 xt x 2 L + c 2 t 2 t L ! cos L xdx n , 1 , 2 , sin x = 1 2 v 0 (0) + X n =1 v n (0) cos L x v n (0) = R L 0 sin x cos L R L 0 cos
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