I am given $u_t$ = $u_{xx} - au$ where a is a nonzero constant, $u(x,0)=f(x)$, $u(0,t)=0$, and $u(1,t)=0$. Solve the time dependent problem via separation of variables method when a>0.
I let $u(x,t)=P(t)Q(x)$ and plugged that into the equation. I got $$ \frac {P'}P=\frac {Q''}Q-a $$ and don't know where to go from here.
Hint
$$ u = X(x)T(t) \implies u_t = T'X,\ u_{xx} = TX'' \implies T'X = TX'' - aXT \implies \frac {T'}T = \frac {X''}X - a $$ LHS is the function of $t$ only, RHS is the function of $x$ only, so this identity to be true, you need to require, that they are the same constant. $$ \frac {T'}T = \frac {X''}X - a = \lambda $$