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 ✗
ClearAll["Global`*"]; pde = -D[s[t, \[Zeta], \[Eta], \[Phi]], t] == ((\[Zeta]^2 - 1)*D[s[t, \[Zeta], \[Eta], \[Phi]], \[Zeta]]^2)/(2*\[Sigma]^2*m*(-\[Eta]^2 + \[Zeta]^2)) + ((-\[Eta]^2 - 1)*D[s[t, \[Zeta], \[Eta], \[Phi]], \[Eta]]^2)/(2*\[Sigma]^2*m*(-\[Eta]^2 + \[Zeta]^2)) + (1*D[s[t, \[Zeta], \[Eta], \[Phi]], \[Phi]]^2)/(2*\[Sigma]^2*m*(\[Zeta]^2 - 1)*(-\[Eta]^2 - 1)) + (a[\[Zeta]] + b[\[Zeta]])/(-\[Eta]^2 + \[Zeta]^2); sol = AbsoluteTiming[TimeConstrained[DSolve[pde, s[t, \[Zeta], \[Eta], \[Phi]], {t, \[Zeta], \[Eta], \[Phi]}], 60*10]];
Failed
Maple ✓
restart; pde := -diff(S(t,xi,eta,phi),t) =1/2*diff(S(t,xi,eta,phi),xi)^2*(xi^2-1)/sigma^2/m/(xi^2-eta^2)+ 1/2*diff(S(t,xi,eta,phi),eta)^2*(1-eta^2)/m/sigma^2/(xi^2-eta^2)+ 1/2*diff(S(t,xi,eta,phi),phi)^2/m/sigma^2/(xi^2-1)/(1-eta^2)+ (a(xi)+b(eta))/(xi^2-eta^2); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,'build')),output='realtime'));
\[S \left (t , \xi , \eta , \phi \right ) = \phi \textit {\_c}_{4}+t \textit {\_c}_{1}+c_{1}+c_{2}+c_{3}+c_{4}-\left (\int \frac {\sqrt {\left (2 \eta ^{2}-2\right ) m \,\sigma ^{2} b \left (\eta \right )-2 \left (\eta -1\right ) \left (\eta +1\right ) \left (\eta ^{2} \textit {\_c}_{1}+\textit {\_c}_{3}\right ) m \,\sigma ^{2}-\textit {\_c}_{4}^{2}}}{\eta ^{2}-1}d \eta \right )-\left (\int \frac {\sqrt {\left (-2 \xi ^{2}+2\right ) m \,\sigma ^{2} a \left (\xi \right )-2 \left (\xi -1\right ) \left (\xi +1\right ) \left (\xi ^{2} \textit {\_c}_{1}+\textit {\_c}_{3}\right ) m \,\sigma ^{2}-\textit {\_c}_{4}^{2}}}{\xi ^{2}-1}d \xi \right )\]