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

Question

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 $0<y<x^2$. 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).

Answer 1

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 $y < x^2$.

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(1) P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM, 1973. doi:10.1137/1.9781611970562.ch1