Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Degasperis Procesi. Solve for \(u(x,t)\) \[ u_t - u_{xxt} + 4 u u_x = 3 u_x u_xx + u u_{xxx} \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[u[x, t], t] - D[D[u[x, t], {x, 2}], t] + 4*u[x, t]*D[u[x, t], x] == 3*D[u[x, t], x]*D[u[x, t], {x, 2}] + u[x, t]*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)-diff(u(x,t),x,x,t)+4*u(x,t)*diff(u(x,t),x)=3*diff(u(x,t),x)*diff(u(x,t),x$2)+u(x,t)*diff(u(x,t),x$3); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t),'build')),output='realtime'));
\[{\it PDESolStruc} \left ( u \left ( x,t \right ) ={\frac {{\it \_F1} \left ( x \right ) }{-{\it \_c}_{{2}}t+{\it \_C2}}},[ \left \{ \left \{ {\it \_F1} \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_a},[ \left \{ {\frac { \left ( {\frac {{\rm d}^{2}}{{\rm d}{{\it \_a}}^{2}}}{\it \_b} \left ( {\it \_a} \right ) \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{\it \_a}+ \left ( {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{\it \_b} \left ( {\it \_a} \right ) {\it \_a}+{\it \_b} \left ( {\it \_a} \right ) \left ( {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) \right ) \left ( {\it \_c}_{{2}}+3\,{\it \_b} \left ( {\it \_a} \right ) \right ) -{\it \_a}\, \left ( {\it \_c}_{{2}}+4\,{\it \_b} \left ( {\it \_a} \right ) \right ) }{{\it \_a}}}=0 \right \} , \left \{ {\it \_a}={\it \_F1} \left ( x \right ) ,{\it \_b} \left ( {\it \_a} \right ) ={\frac {\rm d}{{\rm d}x}}{\it \_F1} \left ( x \right ) \right \} , \left \{ x=\int \! \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{-1}\,{\rm d}{\it \_a}+{\it \_C1},{\it \_F1} \left ( x \right ) ={\it \_a} \right \} ] \right ) \right \} \right \} ] \right ) \] But still has unresolved ODE’s in solution
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