Differential Equations Lecture Work Solutions 156

Differential Equations Lecture Work Solutions 156 - L x dx...

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c. u tt c 2 u xx = xt u ( x, 0) = sin x u t ( x, 0) = 0 u (0 ,t )=0 u x ( L, t )= t w ( x, t )= xt ; w t = x w tt =0 w xx =0 w ( x, 0) = 0 w t ( x, 0) = x v tt c 2 v xx = xt v (0 ,t )= v t ( L, t )=0 λ n = ( n 1 / 2) π L 2 φ n =s in ( n 1 / 2) π L xn =1 , 2 , ··· v ( x, 0) = sin x v t ( x, 0) = x v ( x, t )= X n =1 v n ( t )s in ( n 1 / 2) π L x xt = X n =1 s n ( t )s in ( n 1 / 2) π L x s n ( t )= R L 0 xt sin ( n 1 / 2) π L xdx R L 0 sin 2 ( n 1 / 2) π L xdx v ( x, 0) = sin x = X n =1 v n (0) sin ( n 1 / 2) π L x v n (0) = R L 0 sin x sin ( n 1 / 2) π L xdx R L 0 sin 2 ( n 1 / 2) π L xdx v t ( x, 0) = x = X n =1 ˙ v n (0) sin ( n 1 / 2) π L x ˙ v n (0) = R L 0 x sin ( n 1 / 2) π L xdx R L 0 sin 2 ( n 1
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Unformatted text preview: L x dx v n + ( n 1 / 2) L 2 c 2 v n = s n ( t ) v n = c 1 cos c n t + c 2 sin c n t + Z t s n ( ) sin c n ( t ) c n d v n (0) = c 1 v n (0) = c 2 c n continue as in 3b. 156...
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