- #1
mathwurkz
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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)?
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)?
