Differential Equations Lecture Work Solutions 149

Differential Equations Lecture Work Solutions 149 - 3 utt...

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3. u tt c 2 u xx = S ( x t ) u ( x, 0) = f ( x ) u t ( x, 0) = 0 a. u (0 , t ) = A ( t ) u ( L, t ) = B ( t ) w = α x + β β = A ( t ) αL + β = B α = B β L w = B ( t ) A ( t ) L x + A ( t ) w xx = 0 w tt = ¨ B ¨ A L x + ¨ A v = u w u = v + w v tt c 2 v xx = S ( x, t ) w tt ˆ S ( x, t ) v ( x, 0) = f ( x ) B (0) A (0) L x A (0) F ( x ) v t ( x, 0) = 0 w t ( x, 0) = 0 ˙ B (0) ˙ A (0) L x ˙ A (0) G ( x ) v (0 , t ) = 0 v ( L, t ) = 0 Solve the homogeneous λ n = n π L 2 n = 1 , 2 , · · · φ n = sin n π L x v ( x, t ) = n =1 v n ( t ) sin n π L x ˆ S ( x, t
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