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 )= X n =1 u n ( t ) φ n ( x ) S ( x, t )= X n =1 s n ( t ) φ n ( x ) φ n ( x )=s i n L x λ n = L 2 n =1 , 2 , ··· s n ( t )= R L 0 S ( x, t )s in L xdx R L 0 sin 2 L xdx ¨ u n ( t )+ c 2 L 2 u n ( t )= s n ( t ) u n ( t )= c 1 cos L ct + c 2 sin L ct + Z t 0 s n ( τ ) sin c L ( t τ ) c L u n (0) = c 1 = R L 0 f ( x )s in L xdx R L 0 sin 2 L xdx since u ( x, 0) = f ( x ) ˙ u n (0) = c 2 c
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.

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