So I have to do this:
-\sum_{n=0}^{\infty} \left(\frac{n^2 \pi \ T}{L^2} a_n(t) + m_La^{''}_n\right)\sin\left(\frac{n \pi x}{L} \right) = F_0 \sin \omega_E t.
I'm supposed to multiply this equation through by $\sin(kx\pi/L)$, integrate over $x$ from $0$ to $L$, and use the fact that
$\int_0^L \sin\left(\frac{n \pi x}{L} \right) \sin \left(\frac{k \pi x}{L} \right)dx = \begin{cases}\frac{L}{2}& \mbox{if }n=k\\ 0&\mbox{otherwise} \end{cases}$
to derive a set of ordinary differential equations that model each individual $a_k(t)$.
I tried to do this and got stuck, so I thought I'd come here for help.
This is what I have so far
If $n = k$ -\sum_{n=0}^{\infty} \left ( \int_0^L { \left(\frac{n^2 \pi \ T}{L^2} a_n(t) + m_La^{''}_n\right)}dx \times \frac L 2 \right ) = \int_0^L {F_0 \sin \omega_E t \sin \left(\frac{k \pi x}{L} \right)}dx -\sum_{n=0}^{\infty} \left(\frac{n^2 \pi \ T}{L} a_n(t) + Lm_La^{''}_n \right) \times \frac L 2 = \int_0^L {F_0 \sin \omega_E t \sin \left(\frac{k \pi x}{L} \right)}dx -\sum_{n=0}^{\infty} \left( 2n^2 \pi \ T a_n(t) + 2L^2 m_La^{''}_n \right) = \int_0^L {F_0 \sin \omega_E t \sin \left(\frac{k \pi x}{L} \right)}dx -\sum_{n=0}^{\infty} \left( 2n^2 \pi \ T a_n(t) + 2L^2 m_La^{''}_n \right) = F_0 \sin \omega_E t \left(- \frac{L}{k \pi}\right) \left(\cos \left(k \pi \right) - 1\right) and then i have no idea. If $n \neq k$, then the right side is the same and the left side is $0$.