Differential Equations Lecture Work Solutions 113

Differential Equations Lecture Work Solutions 113 - 5. utt...

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5. u tt c 2 2 u =0 u ( a, θ, t )=0 u ( r, 0 ,t )= u ( r, π/ 2 ,t )=0 ¨ T + λc 2 T =0 Θ 0 + µ Θ=0 r ( rR 0 ) 0 +( λr 2 µ ) R =0 Θ(0) = Θ( π/ 2) = 0 | R (0) | < R ( a )=0 µ n =(2 n ) 2 Θ n =s in2 R ( r )= J 2 n ( λ 2 nm r ) n =1 , 2 , ··· J 2 n ( ± λ 2 n, m a )=0 m =1 , 2 , ··· u ( r, θ, t )= X m =1 X n =1 a mn J 2 n ( ± λ 2 n, m r )s in2 cos c ± λ 2 n, m t | {z } since u t ( r, θ, 0) = 0 g ( r, θ )= u ( r, θ, 0) = X m
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