Create a new moving coordinate system
Translate the derivatives to the new system.
The DE written in the new variables is
Now transform . The new BVP is
Solve this ODE
To have a solution that's bounded at infinity, we must set
Plugging in the BC,
The solution is the inverse Laplace transform of this function. It can be done with convolution.
Everything in the integrand except is called the Green's function for this problem.