I am seeking the explicit formula for the solution of $u_t-\Delta u=cu=f$ in $\mathbb{R}^n$ x $(0,T)$ with initial condition $u(\dot{},0)=g$ and $c \in \mathbb{R}$.
If I let $v(x,t)=e^{-ct}u(x,t)$ and multiply the PDE through by $e^{-ct}$ I get $d_{t}v=u(x,t)e^{-ct}-ce^{-ct}u(x,t)=e^{-ct}\Delta u+e^{-ct}f= \Delta v + e^{-ct}f$. Thus we are left with $d_{t}v-\Delta v= e^{-ct}f$, which is essentially the non homogeneous heat equation with forcing function $ e^{-ct}f$.
Assuming this is the correct change of variables, how do I go about solving this equation. I know the traditional homogeneous equation is solved using separation of variables. I'm just kind of confused how I solve for an explicit solution given the single initial condition I am given. Can I just use the convolution in $\mathbb{R}^n$ of the fundamental solution of the heat equation and the the forcing function $e^{-ct}f$?
Check formula (2) here: http://math.tut.fi/~piche/pde/notes05.pdf or google "Duhamel's principle"