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