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 ✗


Failed

Maple ✓


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