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Question:

Consider the differential equation:

[tex]\frac{d^2 y(x)}{dx^2} + y(x) = f(x) \ \ ; \ \ 0 \leq x \leq L \\[/tex]

The boundard conditions for [tex]y(x)[/tex] are: [tex]y(0) = y(L) = 0 \\[/tex]

Here f(x) is assumed to be a known function that can be expanded in a complete Fourier series:

[tex]f(x) = a_0 + \sum_1^\infty \left[ a_n cos (n \pi x / L ) + b_n \sin (n \pi x / L )\right]\\[/tex]

Write expressions for [tex]a_n[/tex] and [tex]b_n[/tex] Then use the Fourier series to solve for y(x) in the boundary value problem and show that

[tex]y(x) = L^2 \sum_1^\infty \left( \frac{b_n}{L^2 - n^2 \pi ^2}\right) \sin (n \pi x / L ) \\[/tex]

How do I go about finding a_n and b_n so I can solve for y(x) when they do not give f(x)?

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# Differential Equation and Fourier Series

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