Differential Equations Lecture Work Solutions 144

# Differential Equations Lecture Work Solutions 144 - 1a utt...

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1a. u tt c 2 u xx = S ( x, t ) u ( x, 0) = f ( x ) u t ( x, 0) = 0 u (0 , t ) = u ( L, t ) = 0 u ( x, t ) = n =1 u n ( t ) φ n ( x ) S ( x, t ) = n =1 s n ( t ) φ n ( x ) φ n ( x ) = sin n π L x λ n = n π L 2 n = 1 , 2 , · · · s n ( t ) = L 0 S ( x, t ) sin n π L x dx L 0 sin 2 n π L x dx ¨ u n ( t ) + c 2 n π L 2 u n ( t ) = s n ( t ) u n ( t ) = c 1 cos n π L ct + c 2 sin n π L ct + t 0 s n ( τ ) sin c n π L ( t τ ) c n π L u n (0) = c 1 = L 0 f ( x ) sin n π L x dx L 0 sin 2 n π L x dx since u ( x, 0) = f ( x ) ˙ u n (0) = c 2 c n π L = 0
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