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**Deadstar** Solve the heat equation $\displaystyle u_t = u_{xx}$ with the boundary conditions $\displaystyle u(0, t) = u(\pi, t) = 0$ and the following initial condition: $\displaystyle u(x, 0) = T$.

Now, the question itself isn't a problem, its the actual derivation of the heat equation that is (its not asked in the question but we're told that in an exam it would be).

So I'm looking to derive this. $\displaystyle u(x, t) = \sum_1^\infty b_n e^{-n^2t} \sin(nx) $ where $\displaystyle bn = \frac{2}{\pi} \int_0^{\pi} u(x, 0) \sin(nx) dx$.

So i start by doing separation of variables to get $\displaystyle T'X = X''T, ... , X'' = \lambda X $ and $\displaystyle T' = \lambda T$.

Now at this point I think I should use eigenfunctions then superposition to solve it but i cant see where the $\displaystyle e^{-n^2 t}$ or the $\displaystyle b_n$ come from?

Any help please?