#### 2.13.1 Hamilton-Jacobi type PDE

problem number 108

Taken from Maple pdsolve help pages, which is taken from Landau, L.D. and Lifshitz, E.M. Translated by Sykes, J.B. and Bell, J.S. Mechanics. Oxford: Pergamon Press, 1969

Solve for $$S \left ( t,\xi ,\eta ,\phi \right )$$ \begin {align*} -{\frac {\partial }{\partial t}}S \left ( t,\xi ,\eta ,\phi \right ) &=1/2 \,{\frac {\left ( {\frac {\partial }{\partial \xi }}S \left ( t,\xi ,\eta ,\phi \right ) \right ) ^{2} \left ( {\xi }^{2}-1 \right ) }{{\sigma }^{2}m \left ( -{\eta }^{2}+{\xi }^{2} \right ) }}+1/2\,{\frac { \left ( {\frac { \partial }{\partial \eta }}S \left ( t,\xi ,\eta ,\phi \right ) \right ) ^{ 2} \left ( -{\eta }^{2}+1 \right ) }{{\sigma }^{2}m \left ( -{\eta }^{2}+{ \xi }^{2} \right ) }}+1/2\,{\frac { \left ( {\frac {\partial }{\partial \phi }}S \left ( t,\xi ,\eta ,\phi \right ) \right ) ^{2}}{{\sigma }^{2}m \left ( {\xi }^{2}-1 \right ) \left ( -{\eta }^{2}+1 \right ) }}+{\frac {a \left ( \xi \right ) +b \left ( \eta \right ) }{-{\eta }^{2}+{\xi }^{2}}} \end {align*}

Mathematica

Failed

Maple

$S \left (t , \xi , \eta , \phi \right ) = \phi \mathit {\_c}_{4}+t \mathit {\_c}_{1}+c_{1}+c_{2}+c_{3}+c_{4}-\left (\int \frac {\sqrt {\left (2 \eta ^{2}-2\right ) m \sigma ^{2} b \left (\eta \right )-2 \left (\eta -1\right ) \left (\eta +1\right ) \left (\eta ^{2} \mathit {\_c}_{1}+\mathit {\_c}_{3}\right ) m \sigma ^{2}-\mathit {\_c}_{4}^{2}}}{\eta ^{2}-1}d \eta \right )-\left (\int \frac {\sqrt {\left (-2 \xi ^{2}+2\right ) m \sigma ^{2} a \left (\xi \right )-2 \left (\xi -1\right ) \left (\xi +1\right ) \left (\xi ^{2} \mathit {\_c}_{1}+\mathit {\_c}_{3}\right ) m \sigma ^{2}-\mathit {\_c}_{4}^{2}}}{\xi ^{2}-1}d \xi \right )$