Prove that there exists a smooth function $f$ defined on a neighborhood of $(0, 0)$ in $\mathbb{R}^2$ such that $f(0, 0) = 0$ and \begin{align*} \frac{\partial f}{\partial x} &= y e^{-x-y} - f, \\ \frac{\partial f}{\partial y} &= x e^{-x-y} - f. \end{align*}
my attemp:
By integrating the first equation with respect to $x$, I arrived at: $f(x, y) = -y e^{-x-y} + y e^{-y} + g(y) - \int_0^x f \, dx$ where $g(y)$ is a function of $y$ introduced due to the partial integration with respect to $x$.
To proceed, I then differentiated the above with respect to $y$ and using the second equation to equate it to the second equation. However, I encountered challenges when trying to do it since I have two unknow function, now g(y) and the original one.
Can anyone help in elucidating the next steps or provide insights on how to proceed with this system?
Can anyone help in finding an explicit solution for $f(x, y)$ or prove its existence based on the given differential equations and condition?