Given the partial differential equation \begin{equation} (1-x)\left[- f(x,y) + \frac{\partial f(x,y)}{\partial x} \right] + (1-y)\left[- f(x,y) + \frac{\partial f(x,y)}{\partial y} \right] = 0 \end{equation} in the interval $0 \leq x < 1$, $0 \leq y < 1$, the following solution can be obtained by separation of variables: \begin{equation} f(x,y) = C \, \exp(x+y) \end{equation}

Is this the general solution?

Thank you.