Boundary Values & Fourier Series · Challenge problem

A rod $0 \le x \le \pi$ with a uniform heat source obeys $u_t = u_{xx} + 2$, with $u(0,t) = u(\pi, t) = 0$ and $u(x, 0) = 0$. For large $t$, $u\!\left(\dfrac\pi2, t\right) \approx \dfrac{\pi^2}{4} - Ce^{-t}$. Find $C$.