Consider the Heat equation $\frac {\partial u}{\partial t} = k\frac {\partial ^2 u}{\partial t^2}$ subject to: $u(0,t)=u(\frac{\pi}{2},t)=0$. Find the solution that satisfies the initial condition: $u(x,0)=\pi \sin(3x) -5\sin(42x)$.
Having a bit of issue with this problem. I know that the general form of the solution for this will appear as $u(x,t)=\sum_{n=1}^{\infty} B_n e^{-\lambda kt} \sin(\sqrt{\lambda}x)$ and $u(x,0)=\sum_{n=1}^{\infty} B_n \sin(\sqrt{\lambda}x)$. In this case, $\lambda = (2n)^2$.
When I attempt to solve, from the conditions, I get $n =\frac{3}{2}, 21$, respectively, but n is defined as an integer.
Not sure how to continue. Any advice?