I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems.
For example, suppose I have the following terminal value problem:
$$F_t + \frac{1}{2}\sigma^2x^2F_{xx}=1$$
$$F(x,T) = \ln(x)^4,~x>0$$
How would I compute $F(x,t)$ in closed form, given the closed form of the right hand side $(ln(x))^4$ using Feynman-Kac?