Evaluate the constants $\displaystyle b_n$ for the special case $\displaystyle \displaystyle f(x)=\sin\left(\frac{3\pi x}{L}\right)$. On a single graph plot the solutions u(x,t) for $\displaystyle t=0, \ 1, \ 2, \ 3$, assuming $\displaystyle L=\pi, \ k=1$

$\displaystyle \displaystyle u(x,t)=\sum_{n=1}^{\infty}b_n\exp(\lambda_n kt)\sin\left(\frac{n\pi x}{L}\right)$

The book shows to set up solving $\displaystyle b_n$ like:

$\displaystyle \displaystyle\int_0^Lf(x)\sin\left(\frac{m\pi x}{L}\right)dx=\sum_{n=1}^{\infty}b_n\int_0^L\sin\ left(\frac{n\pi x}{L}\right)\sin\left(\frac{m\pi x}{L}\right)$

Now, is the f(x) in how to setup finding b_n the f(x) in the problem?

Thanks.