Partial Differential Equations

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:confused::confused:

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

Then it wants the general solution for X(x) and T(t)

and then assuming $\displaystyle D>0, a>=0, L>0$ solve the boundary value problem:

$\displaystyle u_t - Du_{xx} - au = 0$ for all $\displaystyle 0=<x=<L , t>=0$

$\displaystyle u(0,t) = u(L,t) = 0 $ for all $\displaystyle t>=0$

$\displaystyle

u(x,0) = sin(\pi x/L) + sin(2\pi x/L)

$

and finally characterise the differenc between solutions for $\displaystyle a>D\pi^2 / L^2$ and $\displaystyle a<D\pi^2 / L^2$

I have found a similiar problem in Boyce's elementary Diff equations book but just cant work it out!

Thanks