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'));
\[u \left (x , t\right )=\frac {\textit {\_F1} \left (x \right )}{-\textit {\_c}_{2} t +c_{2}}\boldsymbol {\mathrm {where}}\left [\left \{\left \{\textit {\_F1} \left (x \right )=\textit {\_a} \boldsymbol {\mathrm {where}}\left [\left \{\frac {\textit {\_a} \textit {\_}b\left (\textit {\_a} \right )^{2} \left (\frac {d^{2}}{d \textit {\_a}^{2}}\mathrm {\_}\mathrm {b}\left (\textit {\_a} \right )\right )+\textit {\_a} \textit {\_}b\left (\textit {\_a} \right ) \left (\frac {d}{d \textit {\_a}}\mathrm {\_}\mathrm {b}\left (\textit {\_a} \right )\right )^{2}+\left (\textit {\_c}_{2}+3 \textit {\_}b\left (\textit {\_a} \right )\right ) \textit {\_}b\left (\textit {\_a} \right ) \left (\frac {d}{d \textit {\_a}}\mathrm {\_}\mathrm {b}\left (\textit {\_a} \right )\right )-\left (\textit {\_c}_{2}+4 \textit {\_}b\left (\textit {\_a} \right )\right ) \textit {\_a}}{\textit {\_a}}=0\right \}, \left \{\textit {\_a} =\textit {\_F1} \left (x \right ), \textit {\_}b\left (\textit {\_a} \right )=\frac {d}{d x}\textit {\_F1} \left (x \right )\right \}, \left \{x =c_{1}+\int \frac {1}{\textit {\_}b\left (\textit {\_a} \right )}d \textit {\_a} , \textit {\_F1} \left (x \right )=\textit {\_a} \right \}\right ]\right \}\right \}\right ]\] But still has unresolved ODE’s in solution
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