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:

$F_t+\frac{1}{2}σ^2x^2F_{xx}=1$

$F(x,T)=(ln(x))^4,\, x>0$

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?

Thanks in advance,

Sriram