Uniqueness of solutions of quasilinear equation $(y+u)\frac{\partial u}{\partial x} +y\frac{\partial u}{\partial y} = x-y$

Question

Uniqueness of quasilinear equation $(y+u)\frac{\partial u}{\partial x} +y\frac{\partial u}{\partial y} = x-y$ with initial condition $u(x,1)=1+x$. Prove that the solution is unique in $C^1 (\Omega)$, where $\Omega = \{(x,y) \in \mathbb R^2, y>0 \}$. I tried method of characteristics to solve it explicitly but failed since it seems quite tricky. Instead, I found the link(http://nptel.ac.in/courses/111103021/7.pdf), where Theorem 6 gives the uniqueness condition and I checked and it works here, but I doubt it.

Answer 1

Solving the PDE in order to only prove the uniqueness of solution isn't a smart method. But it works anyway.

The method of characteristics is not tricky at all (see below). It leads to a unique solution : $$u(x,y)=\frac{2}{y}+x-y$$