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Differential Equations Lecture Work Solutions 70

# Differential Equations Lecture Work Solutions 70 - n=1(An...

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+ n =1 ( A n b n + B n b n ) b n δ n sin n θ These are Fourier series of f ( θ ) and g ( θ ) thus the coeﬃcients α 0 , α n , β n for f and the coeﬃcients γ 0 , γ n , δ n for g can be written as follows α 0 = 1 2 π 2 π 0 f ( θ ) α n = 1 π 2 π 0 f ( θ ) cos nθ dθ β n = 1 π 2 π 0 f ( θ ) sin nθ dθ γ 0 = 1 2 π 2 π 0 g ( θ ) γ n = 1 π 2 π 0 g ( θ ) cos nθ dθ δ n = 1 π 2 π 0 g ( θ ) sin 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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