Differential Equations Lecture Work Solutions 76

Differential Equations Lecture Work Solutions 76 - n=1 n x=...

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X n =1 b 0 n ( t )s in L x = X n =1 b 0 n ( t ) " L ± 2 # sin L x + X n =1 q n ( t )s in We have a Fourier Sine series on left and Fourier Sine series on right, so the coefficients must be the same; i.e., (a) b 0 n ( t )= L 2 b n ( t )+ q n ( t ) A ±rst order ODE for b n ( t ). III. Solve b 0 n ( t )= L 2 b n ( t )+ q n ( t ) Solution Form: b n ( t )= A n b H n ( t )+ b P n ( t ) Homogeneous Solution: b H n ( t )= e ( L ) 2 t Particular Solution: b P n ( t )= e ( L ) 2 t Z t 0 e ( L ) 2 τ q n ( τ ) b n ( t )= A n e ( L ) 2 t + e ( L ) 2 t Z t 0 e ( L ) 2 τ q n ( τ ) (Step IV is an extra step, not required in homework problem.) IV .F ind A n from initial condition. u ( x, 0) = f ( x )= X n =1 b n (0) sin L x b n (0) =
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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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