Given two differential basis vectors find the change of coordintes

Question

I'm given that $\partial_u = ye^{-x} \partial_x, \partial_v = y\partial_x +y^2 \partial_y$ and asked to find $u(x,y), v(x,y)$. I thought maybe something like a Jacobian would be appropriate but representing the given information as a matrix equation I would have $\frac{\partial u}{\partial x} = ye^{-x}$ and $\frac{\partial u}{\partial y} =0$ which cannot be consistently solved. How do I proceed?