Differential Equations Lecture Work Solutions 65

Differential Equations Lecture Work Solutions 65 - 7 a. urr...

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7a. u rr + 1 r u r + 1 r 2 u θθ =0 u θ ( r, 0) = 0 u ( r, π/ 2) = 0 u (1 )= f ( θ ) r 2 R 0 + rR 0 λR Θ 0 + λ Θ = 0 no periodicity !! Θ 0 (0) = 0 R n = c n r 2 n 1 + D n r 2 n 1 Θ 0 ( π/ 2) = 0 boundedness implies R n = c n r 2 n 1 If λ< 0 trivial λ Θ 0 = A 0 θ + B 0 Θ 0 0 = A 0 Θ 0 0 (0) = 0 A 0 Θ 0 ( 2) = 0 B 0 trivial λ> 0 Θ= A cos λθ + B sin Θ 0 = λA sin λθ + B λ cos Θ 0 (0) = 0 B Θ( 2) = 0 A cos λπ/ 2=0 2= n 1 2 πn =1 , 2 , ··· λ =2 n 1 2 n 1
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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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