Solve the equation $$u_t=17u_{xx}, \ 0<x<\pi, \ t>0,$$ with the boundary conditions $$u(0,t)=u(\pi,t)=0, \ t\ge 0,$$ and the initial conditions $$u(x,0)=\left\{ \begin{array}{l l} 0 & \quad \text{if} \ 0\le x\le \pi/2\\ 2 & \quad \text{if} \ \pi/2<x\le\pi \end{array} \right.$$
How will I be able to solve this PDE? My book didn't provide an example of how to solve a PDE of the form $u_t=Au_{xx}$ and I am not sure on how to go about solving it?
Consider $u(x,t)=X(x)T(t)$. Then $u_{xx}=X''(x)T(t)$ and $u_t=X(x)T'(t)$. Substitute in the original equation to get
$X(x)T'(t)=17X''(x)T(t).$ So
$\dfrac{T'(t)}{T(t)}=17\dfrac{X''(x)}{X(x)}=-\lambda$. So this gives
$T'(t)+\lambda T=0$ and $X''(x)+\dfrac{\lambda}{17}X=0.$ Can you proceed from here. I think its trivial now.