Hi I am stuck on a PDE boundary question, I think I have shown the equation seperates and understand what the boundarys mean but I am not sure how to show the general solution and then solve the boundary value problems. Please help

Assume solution $\displaystyle u(x,t) = X(x)T(t)$ to the modified diffusion equation $\displaystyle u_t - Du_{xx} - au = 0$.

First it says show the equation seperates, I think the follwing is correct?:

substituting the solution into the equation:

$\displaystyle u_t - au = Du_{xx}$

$\displaystyle

XT' - aXT = DX''T

$

so $\displaystyle \frac {T'}{T} - a = D \frac {X''}{X}$