Differential Equations Lecture Work Solutions 146

Differential Equations Lecture Work Solutions 146 - 2. utt...

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2. u tt c 2 u xx + βu t =co s wt u ( x, 0) = f ( x ) u t ( x, 0) = 0 u (0 ,t )= u ( π, t )=0 φ n =s in nx n =1 , 2 , ··· λ n = n 2 u ( x, t )= X n =1 u n ( t )s in nx cos wt = X n =1 s n ( t )s in nx s n ( t )= cos wt R π 0 sin nx dx R π 0 sin 2 nx dx = A n cos wt X n =1 u n + c 2 n 2 u n + β ˙ u n )s in nx = X n =1 s n ( t )s in nx (*) ¨ u n + β ˙ u n + c 2 n 2 u n = s n ( t )= A n cos wt For the homogeneous: Let u n = e µt ( µ 2 + βµ + c 2 n 2 )=0 µ = β ± β 2 4 c 2 n 2 2 For β< 2 c, β 2 4 c 2 n 2 < 0 complex conjugate roots u n = c 1 cos 4 c 2 n 2 β 2 2 t + c 2 sin 4 c 2 n 2 β 2 2 ± e (
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