# Singularity of Hyperbolic PDE $u_x+2 (x+\sqrt{x^2-y}) u_y=0$

I am studying first order hyperbolic PDEs and want to know if there is a technique to find singularities in your solution without explicitly solving. The PDE in particular is $$u_x+2\left(x+\sqrt{x^2-y}\right) u_y=0$$ where $$0. Any thoughts or resources? (On a side note, Mathematica was able to produce a solution: $$u(x,y)=F(\frac12 \log{(-1 - 2 x - 2 x^2 + 2 (1 + x) \sqrt{x^2 - y} + y)})$$, where $$F$$ is your boundary condition. Any thought of techniques to solve? MOC yielded me a nonlinear ODE that I couldn't solve, so it appears to not be useful here).

Assume a singularity for which $$q = u_y$$ becomes infinite (1). Differentiating the PDE w.r.t. $$y$$, we have: $$q_x + 2 \left( x + \sqrt{x^2 - y} \right) q_y = \frac{q}{\sqrt{x^2 - y}} \, .$$ Thus, along the PDE's characteristic curves $$s\mapsto (x(s), y(s))$$, the unknown $$u$$ is constant and we have $$\dot q(s) = \frac{q(s)}{\sqrt{x(s)^2 - y(s)}} \, .$$ Therefore, if the boundary data is smooth, then the method of characteristics provides smooth solutions everywhere inside the domain where $$y < x^2$$.