I have the following problem:
Solve $\dfrac{\partial u}{\partial t} = \dfrac{1}{r}\dfrac{\partial^2 (ru)}{\partial r^2}, r \le a$ subject to:
$u(r,0)=u_0$ and $\dfrac{1}{k}\dfrac{\partial u}{\partial r} = u_0 - u(a,t) - t$ at $r=a$
I would like to see how this was solved using the Laplace transform, but am struggling to solve the resulting ODE.