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'));
\[S \left ( t,\xi ,\eta ,\phi \right ) ={\it \_c}_{{1}}t+{\it \_c}_{{4}}\phi -\int \!{\frac {\sqrt { \left ( -2\,{\xi }^{2}+2 \right ) m{\sigma }^{2}a \left ( \xi \right ) -2\,m \left ( \xi -1 \right ) \left ( \xi +1 \right ) \left ( {\xi }^{2}{\it \_c}_{{1}}+{\it \_c}_{{3}} \right ) {\sigma }^{2}-{{\it \_c}_{{4}}}^{2}}}{{\xi }^{2}-1}}\,{\rm d}\xi -\int \!{\frac {\sqrt { \left ( 2\,{\eta }^{2}-2 \right ) m{\sigma }^{2}b \left ( \eta \right ) -2\,m \left ( \eta -1 \right ) \left ( \eta +1 \right ) \left ( {\eta }^{2}{\it \_c}_{{1}}+{\it \_c}_{{3}} \right ) {\sigma }^{2}-{{\it \_c}_{{4}}}^{2}}}{{\eta }^{2}-1}}\,{\rm d}\eta +{\it \_C1}+{\it \_C2}+{\it \_C3}+{\it \_C4}\]