In my course on PDEs, I was given this PDE similar to the heat equation, which we solve for $t \geq 0$ and $0 \leq x \leq 1$
$u_t=\alpha u_{xx} + u$ with $\alpha$ a positive constant. The conditions provided are $u_x(0,t)=u_x(1,t)=0$ and $u(x,0)=\sin^2(\pi x)$.
I was not told how to solve this equation. This looks similar to the heat equation which I know can be solved using separation of variables. I am not very familiar with separation of variables or Fourier series, so I was wondering, can this be solved with separation of variables? I found myself unable to solve this equation using separation of variables, probably due to lack of familiarity. Can anyone please show me how to solve this? I thank all helpers.