Q:

PDE of the form $x \partial_x T - y \partial_y T = F(x,y)$ where $F$ is a given function.

Is there a known solution, or technique, for solving the following PDE?
$x \partial_x T - y \partial_y T = F(x,y)$
Here, $F$ is a given smooth function $\mathbb R^2 \to \mathbb R$, and $T: \mathbb R^2 \to \mathbb R$ is the solution we seek.
Example. If $F$ is a constant function with value $c$ , a possible solution is
$T = a \log|x| - b\log|y|$
for $a+b = c$.
Singularities. I expect there to be singularities at least at the origin, and also along one of the coordinate axes. 
 Boundary conditions. I don't know if this helps, but I am mainly considering the case when $F = A(x)B(y)$ is a product of two functions in each variable, each of $A$ and $B$ having the property that it is a monotone increasing function taking the value $A(x) = 0$ for $x \leq 0$, and equaling 1 for $x$ large enough. If it makes things even simpler, I'm also fine taking $A(t) = B(t)$. This concretely implies that $F$ is some non-negative function on the first quadrant, and $F$ is zero on all other quadrants.

A:

This is a quasilinear PDE, and it can be solved using the method of characteristics.

