Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Dym equation. Solve for \(u(x,t)\) \[ u_t =u^3 u_{xxx} \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[u[x, t], t] == u[x, t]^3*D[u[x, t], {x, 3}]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
Failed
Maple ✓
restart; pde := diff(u(x,t),t)=u(x,t)^3 * diff(u(x,t),x$3); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t),'build')),output='realtime'));
\[u \left ( x,t \right ) ={\frac {1}{\sqrt [3]{-3\,{\it \_c}_{{1}}t+{\it \_C4}}}\RootOf \left ( -\int ^{{\it \_Z}}\! \left ( \RootOf \left ( -\ln \left ( {\it \_f} \right ) +2\,\int ^{{\it \_Z}}\!{\frac {{\it \_h}}{2\,\sqrt [3]{2}\sqrt [3]{-{{\it \_c}_{{1}}}^{2}}\RootOf \left ( \AiryBi \left ( {\it \_Z} \right ) \sqrt [3]{2}\sqrt [3]{-{{\it \_c}_{{1}}}^{2}}{\it \_C1}\,{\it \_h}+\sqrt [3]{2}\sqrt [3]{-{{\it \_c}_{{1}}}^{2}}{\it \_h}\,\AiryAi \left ( {\it \_Z} \right ) +2\,\AiryBi \left ( 1,{\it \_Z} \right ) {\it \_C1}\,{\it \_c}_{{1}}+2\,\AiryAi \left ( 1,{\it \_Z} \right ) {\it \_c}_{{1}} \right ) +{{\it \_h}}^{2}}}{d{\it \_h}}+{\it \_C2} \right ) \right ) ^{-1}{d{\it \_f}}+x+{\it \_C3} \right ) }\] has RootOf
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