Differential Equations Lecture Work Solutions 70

Differential Equations Lecture Work Solutions 70 - + n=1...

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+ X n =1 ( A n b n + B n b n ) b n | {z } δ n sin These are Fourier series of f ( θ )and g ( θ ) thus the coefficients α 0 n n for f and the coefficients γ 0 n n for g can be written as follows α 0 = 1 2 π Z 2 π 0 f ( θ ) α n = 1 π Z 2 π 0 f ( θ )cos nθ dθ β n = 1 π Z 2 π 0 f ( θ )s in nθ dθ γ 0 = 1 2 π Z 2 π 0 g ( θ ) γ n = 1 π Z 2 π 0 g ( θ )cos nθ dθ δ n = 1 π Z 2 π 0 g ( θ )s in nθ dθ On the other hand these coefficients are related to the unknowns A 0 ,a 0 ,B 0 ,b 0 ,A n ,a n ,B n and b n via the three systems of 2 equations each α 0 = A 0 a 0 + B 0 a 0 ln a γ 0 = A 0 a 0 + B 0 a 0 ln b solve for A 0 a 0 ,B 0 a 0 α n =( A n a n + B n a n ) a n γ n =( A n b n + B n b n ) a n
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