Method of Separation of Variables

Question

How can I deduce $C_n=\int_0^L f(x)\sin(\frac{n\pi}{L}x)\,dx$

from

$\displaystyle\sum_{n=1}^{\infty}C_n\sin(\frac{n\pi}{L}x)=f(x)$,
where $0<x<L$

Answer 1

You can't, because you are off by a constant factor. To find the value of $C_n$, multiply both sides by $\sin{\left (m \pi x/L \right )}$ and integrate with respect to $x$:

$$\sum_{n=0}^{\infty} C_n \int_0^L dx \, \sin{\frac{m \pi x}{L}} \, \sin{\frac{n \pi x}{L}} = \int_0^L dx \, f(x) \sin{\frac{m \pi x}{L}}$$

Note that one (i.e., you) may derive that

$$\int_0^L dx \, \sin{\frac{m \pi x}{L}} \, \sin{\frac{n \pi x}{L}} = \begin{cases} 0 & m \ne n \\ \frac{L}{2} & m = n \end{cases}$$

The integral thus acts as a filter, and the sum reduces to a simple algebraic expression for $C_m$:

$$C_m = \frac{2}{L} \int_0^L dx \, f(x) \sin{\frac{m \pi x}{L}}$$