Fundamental Solution for Nonhomogeneous Heat Equation?

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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.