So, I need some help to move forward with this problem.
$$ \begin{cases} \frac{\partial F(t,x,y)}{\partial t}+\frac{1}{2}\frac{\partial^2 F(t,x,y)}{\partial x^2}+\frac{9}{2}\frac{\partial^2 F(t,x,y)}{\partial y^2}+\frac{\partial^2 F(t,x,y)}{\partial x \partial y}-F(t,x,y)=0 \ \ \ (t,x) \in[0,T) x R^2\\ F(T,x,y)=x^2y\\ \end{cases} $$
I have concluded that the C matrix defined as $\sigma*\sigma^T=\begin{matrix} 1 & 1\\ 1 & 3 \end{matrix}$
and that $\mu_{1}=\mu_{2}=0$
and that I should solve $$F(t,x,y)=E_{t,x,y}[e^{-(T-t)}x^2y]$$
but my problem is to figure out how $x$ and $y$ should look like. My attempt was $$dX(s)=dW_{1}(s)+dW_{2}(s)$$ $$dY(s)=3dW_{2}(s)+dW_{1}(s)$$ since they should be correlated given the problem, but I'm unsure if I have understood how to define $dX(s)$ and $dY(s)$ properly.
Well, if i preceed with $dX(s)$ and $dY(s)$ as defined above I get
$$F(t,x,y)=E_{t,x,y}[e^{-(T-t)}x^2y]=\\e^{-(T-t)}E_{t,x,y}\bigg[\bigg(x+\big(W_{1}(T)-W_{1}(t)\big)+\big(W_{2}(T)-W_{2}(t)\big)\bigg)^2\bigg(y+3\big(W_{2}(T)-W_{2}(t)\big)+\big(W_{1}(T)-W_{1}(t)\big)\bigg)\bigg]$$
which does not lead me to the correct answer. Anyone with some guiding for me?