Separation of variables PDE problem

Question

Use separation of variables to solve the initial boundary value problem: $$u_t=u_{xx}+2u \cos(t)\quad\text{on}\quad x \in (0,\pi),\quad t>0,$$ $$u(0,t)=u(\pi,t)=0,$$ $$u(x,0)=\phi(x).$$

I look up this old post too get an idea, but I am still not sure how to treat the $2u \cos(t)$ part. PDE separation of variables

Answer 1

Let $u=T(t)X(x)$. If you can get to

$ \displaystyle \frac{T_t}{T}-2\cos t = \frac{X_{xx}}{X}$

then both sides of the equation must be equal to the same constant $C$, and you can take it from there.