2.11.1 \(u_{\theta \theta }+\frac {v^2}{1-\frac {v^2}{c^2}} u_{vv} + v u_v=0\)

problem number 106

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'));
 

\[u \left ( \theta ,v \right ) ={\frac { \left ( {{\rm e}^{\sqrt {{\it \_c}_{{1}}}\theta }} \right ) ^{2}{\it \_C1}+{\it \_C2}}{{{\rm e}^{\sqrt {{\it \_c}_{{1}}}\theta }}v}{{\rm e}^{1/4\,{\frac {{v}^{2}}{{c}^{2}}}}} \left ( \WhittakerM \left ( -1/2\,{\it \_c}_{{1}}+1/2,i/2\sqrt {{\it \_c}_{{1}}},1/2\,{\frac {{v}^{2}}{{c}^{2}}} \right ) {\it \_C3}+ \WhittakerW \left ( -1/2\,{\it \_c}_{{1}}+1/2,i/2\sqrt {{\it \_c}_{{1}}},1/2\,{\frac {{v}^{2}}{{c}^{2}}} \right ) {\it \_C4} \right ) }\]