PDE with terminal condition, Feynman-Kac

Question

I have the following PDE: $$2u_t+9u_{xx}-2u_x=0$$ $$u(x,T)=e^x$$ and I get that $$dX(s)=-dS+3dW(s)$$ $$X(t)=x$$

But how do I get the expected value of $e^x$? I tried substituting $Y(s)=e^x$, but I get the wrong answer... The answer should be $u(x,t)=e^{(x+\frac{7}{2}(T-t))}$.

Answer 1

Hint:

Your SDE results in the following expression:

$X(T) = x - (T - t) + 3(W(T) - W(t))$

Now you just need to find $E[e^{X(T)}]$ Can you do that?

Hint2: MGF