TITLE: How is a heat partial differential equation with a flux term
QUESTION [5 upvotes]: Consider a metal rod, with $K=1$ and longitude $\pi$ .
On its extremes, there is heat transmission with the ambient, according to the differential equation:
$$u_t(x,t)=u_{xx}(x,t)-hu(x,t),$$
with $h>0$.
The extremes of the rod have fixed temperatures of $0^\circ C$ on the left, and $1^\circ C$ on the right. The initial temperature of the rod is the function:
$$f(x)=x(\pi-x).$$
Find the temperature distribution $u(x,t)$ on the rod.
Could someone please explain me how to start? I've never seen something like that (with an $hu(x,t)$ added or taken from the equation). I'm really stuck.
Thank you beforehand, and sorry for my english, it's not my natural language.

REPLY [3 votes]: It sounds like you know how to solve the "usual" heat equation, $u_t=u_{xx}$ by separation of variables. Is that correct? If so, proceed in exactly the same way as you did before, i.e. letting $u(x,t)=X(x)T(t)$. 
The eigenvalue problem will be just like before, but the temporal problem will look like ${T'\over T}+h=-\lambda$.