Added June 20, 2019 From https://en.wikipedia.org/wiki/Chaplygin%27s_equation
Solve for \(u(\theta ,v)\) \[ u_{\theta \theta }+\frac {v^2}{1-\frac {v^2}{c^2}} u_{vv} + v u_v=0 \] Here \(c\) is the speed of sound.
Mathematica ✗
ClearAll["Global`*"]; pde = D[u[theta, v], {theta, 2}] + v^2/(1-v^2/c^2)* D[u[theta,v],{v,2}]+v*D[u[theta,v],v]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[theta, v], {theta, v}, Assumptions->c>0], 60*10]];
Failed
Maple ✓
restart; pde := diff(u(theta,v),theta$2)+ v^2/(1-v^2/c^2)* diff(u(theta,v),v$2)+v*diff(u(theta,v),v)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(theta,v),'build') assuming c>0),output='realtime'));
\[u \left (\theta , v\right ) = \frac {\left (c_{1} {\mathrm e}^{2 \theta \sqrt {\mathit {\_c}_{1}}}+c_{2}\right ) \left (c_{3} \WhittakerM \left (-\frac {\mathit {\_c}_{1}}{2}+\frac {1}{2}, \frac {i \sqrt {\mathit {\_c}_{1}}}{2}, \frac {v^{2}}{2 c^{2}}\right )+c_{4} \WhittakerW \left (-\frac {\mathit {\_c}_{1}}{2}+\frac {1}{2}, \frac {i \sqrt {\mathit {\_c}_{1}}}{2}, \frac {v^{2}}{2 c^{2}}\right )\right ) {\mathrm e}^{\frac {v^{2}}{4 c^{2}}} {\mathrm e}^{-\theta \sqrt {\mathit {\_c}_{1}}}}{v}\]