# Differential Equation and Fourier Series

1. Dec 11, 2005

### mathwurkz

I have this problem. I would appreciate it if anyone can help me get started.
Question:
Consider the differential equation:
$$\frac{d^2 y(x)}{dx^2} + y(x) = f(x) \ \ ; \ \ 0 \leq x \leq L \\$$
The boundard conditions for $$y(x)$$ are: $$y(0) = y(L) = 0 \\$$
Here f(x) is assumed to be a known function that can be expanded in a complete Fourier series:
$$f(x) = a_0 + \sum_1^\infty \left[ a_n cos (n \pi x / L ) + b_n \sin (n \pi x / L )\right]\\$$
Write expressions for $$a_n$$ and $$b_n$$ Then use the Fourier series to solve for y(x) in the boundary value problem and show that
$$y(x) = L^2 \sum_1^\infty \left( \frac{b_n}{L^2 - n^2 \pi ^2}\right) \sin (n \pi x / L ) \\$$
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)?

2. Dec 11, 2005

### Galileo

Well, you can't.Obviously.

Hopefully, you will be able to write down the expressions for them.

3. Dec 11, 2005

### mathwurkz

Ok. I got it now thnks.