4. [20 points] Find the solution to the following IBVP
$\begin{cases}
u_t = u_{xx} + \sin(3\pi x), & 0 < x < 1, t > 0 \
u(0, t) = 0, & t > 0 \
u(1, t) = 0, & t > 0 \
u(x, 0) = \sin(\pi x), & 0 < x < 1,
\end{cases}$ (4)
Hint: As you solve this problem, check for singularities in the coefficients in your various eigen-
function expansions. This will lead to different cases for the initial value problems that
$T_n(t)$
satisfy!