We are told that the solution to the boundary value problem
$$ \frac{ \partial F}{\partial t}(t,x) + \mu (t,x) \frac{\partial F}{ \partial x}(t,x) + \frac{1}{2}\sigma^{2}(t,x)\frac{\partial ^2 F}{\partial x^{2}}(t,x) = 0 \ \ ,\ \ \ \ \ \ \ 0 \leq t \leq T \\ F(T,x) = \phi (x)$$
has the following stochastic representation:
$$ F(t,x) = \mathbb{E}[\phi (X_{T})|X_{t} = x] $$
where $\{X_{t}\}_{t \geq 0}$ is the solution to the SDE
$$ dX_t = \mu(t, X_t)dt+\sigma(t,X_t)dW_t \ , \ \ \ \ \ 0 \leq t \leq T, $$
Then we're asked to deduce that the solution to the boundary value problem
$$ \frac{ \partial F}{\partial t}(t,x) + \mu (t,x) \frac{\partial F}{ \partial x}(t,x) + \frac{1}{2}\sigma^{2}(t,x)\frac{\partial ^2 F}{\partial x^{2}}(t,x) - V(x,t)F(x,t)+f(x,t) = 0 \ \ ,\ \ \ \ \ \ \ 0 \leq t \leq T \\ F(T,x) = \phi (x)$$
Is given as
$$ F(t,x) = \mathbb{E}[ \int_t^Te^{-\int_t^rV(X_u,u)du}f(X_r,r)dr+e^{-\int_t^TV(X_u,u)du} \phi (X_{T})|X_{t} = x] $$
where $\{X_{t}\}_{t \geq 0}$ is as before.
I'm really not sure how to go about doing this. I don't have much of a differential equations background, so it might be something that is quite basic if you know about DEs techniques. Any help would be much appreciated!
This is a generalization of the Feynman-Kac functional.
Here's a hint on how to proceed. First, consider the system $$dX_t = \mu(t,X_t)dt +\sigma(t,X_t)dW_t$$ $$dZ_t = -V(t,X_t)Z_tdt$$
Find the generator of the joint process $(X_t,Z_t)$. Apply the generator to $F(t,X(t))Z(t)$ and use Ito's rule to find an expression for $\mathbb{E}\left[F(T,X(T))Z(T)\middle|...\right]-F(0,X(0)Z(0)$. This will give the answer.
You can also add in a levy process for a stronger result.
For reference, check out "Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation" Chapter 7, Floyd Hanson.
Or, "Stochastic Calculus, Applications in Science and Engineering" Chapter 6, Mircea Grigoriu.
Lots of stuff in the finance literature too.