Problem $3\mathrm{.}1($Problem $3\mathrm{.}1\mathrm{.}5,$ modified$).$ Consider the damped heat equation

$\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}=\frac{de{l}^{2}u}{del{x}^2}-u$

where $>0$ is a constant.

a$)$ Find all real$-$valued solutions of the form

$u(x,t)=e^{t}v(x)$

b$)$ Which of these solutions satisfy the periodic boundary conditions $u(-,t)=u(,t),\frac{du}{dx}(-,t)=\frac{du}{dx}(,t)?$

c$)$ Find the series solution to the IBVP

$($Note: this includes finding a formula for the coefficients.$)$