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

problem number 105

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'));
 

\[u \left (\theta , v\right ) = \frac {{\mathrm e}^{\frac {-4 \sqrt {\textit {\_c}_{1}}\, \theta \,c^{2}+v^{2}}{4 c^{2}}} \left (\operatorname {WhittakerW}\left (-\frac {\textit {\_c}_{1}}{2}+\frac {1}{2}, \frac {i \sqrt {\textit {\_c}_{1}}}{2}, \frac {v^{2}}{2 c^{2}}\right ) c_{4} +\operatorname {WhittakerM}\left (-\frac {\textit {\_c}_{1}}{2}+\frac {1}{2}, \frac {i \sqrt {\textit {\_c}_{1}}}{2}, \frac {v^{2}}{2 c^{2}}\right ) c_{3} \right ) \left (c_{1} {\mathrm e}^{2 \sqrt {\textit {\_c}_{1}}\, \theta }+c_{2} \right )}{v}\]