Fundamental Solution for Nonhomogeneous Heat Equation?

[µ³]

Homework Statement

So I'm trying to solve Evans - PDE 2.5 # 12...

"Write down an explicit formula for a solution of

u_t - \Delta u + cu=f with (x,t) \in R^n \times (0,\infty)
u(x,0)=g(x)"

Homework Equations

The Attempt at a Solution

I figure if I can a fundamental solution for

u_t - \Delta u + cu = 0,

The rest follows through straight forward. I've tried multiplying the solution to the heat equation by a number of terms such as e^{-ct}, e^{v(x,t)}, but everything I've tried so far either gives me a non-fundamental solution, or a non-linear pde.

I've also tried mimicking the two ways they give to finding the solution to the original equation but neither seem to work. Looking for solutions of the form u(x,t) = v(x^2/t) and looking for solutions of the form u(x,t) = 1/t^\beta v(x/t^\alpha) but neither seems to reduce the equation in the problem to a single variables

Any hints?

 

[µ³]

yay, figured it out.

letting u(x,t)=e^{-ct}v(x,t), with v(x,t) solving

v_t - \Delta v = f(x,t) e^{ct}
v(x,0) = g(x)

Solves the original equation. I guess I was just thinking too hard.