1D Heat Equation, Non-Insulated Infinite Bar

Question

Using Fourier Transform in $x$, find the solution $ u(x,t) $ to the PDE

$ u_t = c^2u_{xx} - hu$

with the initial conditions

$ u(x,0) = e^{-x^2} (-\infty < x< \infty)$

Could someone guide me through how to set up this problem?

Answer 1

Make the change $v=e^{ht}u$ and get the heat equation for $v$.