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:

\begin{align} & F_t+\frac{1}{2}σ^2x^2F_{xx}=1\\ & F(x,T)=(\ln(x))^4,\ x>0 \end{align}

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?

Thanks in advance,

Sriram