Differential Equations Lecture Work Solutions 111

Differential Equations Lecture Work Solutions 111 - 4. utt...

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4. u tt c 2 2 u =0 u r ( a, θ, t )=0 u ( r, θ, 0) = 0 u t ( r, θ, 0) = β ( r )cos5 θ ¨ T + λc 2 T =0 Θ 0 + µ Θ=0 r ( rR 0 ) 0 +( λr 2 µ ) R =0 T (0) = 0 Θ (0) = Θ (2 π ) | R (0) | < Θ 0 (0) = Θ 0 (2 π ) R 0 ( a )=0 T = a cos c λ nm t + b sin c λ nm 0 =0 Θ 0 =1 R = J n ( λr ) Since T (0) = 0 R 0 ( a )= J 0 n ( λa ) · λ =0 µ n = n 2 Θ m = cos sin ⇓⇓ T =s in c λ nm n 0 =0 or J 0 n ( λ nm a )=0 m =1 , 2 , ··· for each n =0 , 1 , 2 , ··· u ( r, θ, t )= X m =0 X n =0 { a nm cos + b nm sin } ± J n ( ² λ nm r ) ³ sin c ² λ nm t u t ( r, θ, 0) = X m =0 X n =0 { a nm cos + b nm sin } J n ( ² λ nm r )
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