2.15.12 Dym equation $u_t=u^3{u}_{xxx}$

problem number 121

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'));
 

$$u(x,t)=\mathrm{RootOf}(-{\int }^{\mathit{\_}\mathit{Z}}{\left(\mathrm{RootOf}(-\ln \left(\mathit{\_}\mathit{f}\right)+2{\int }^{\mathit{\_}\mathit{Z}}\frac{\mathit{\_}\mathit{h}}{2\sqrt[3]{2}\sqrt[3]{-{{\mathit{\_}\mathit{c}}_1}^2}\mathrm{RootOf}(\sqrt[3]{2}\sqrt[3]{-{{\mathit{\_}\mathit{c}}_1}^2}\mathrm{AiryBi}\left(\mathit{\_}\mathit{Z}\right)\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{h}+\sqrt[3]{2}\sqrt[3]{-{{\mathit{\_}\mathit{c}}_1}^2}\mathit{\_}\mathit{h}\mathrm{AiryAi}\left(\mathit{\_}\mathit{Z}\right)+2\mathrm{AiryBi}(1,\mathit{\_}\mathit{Z})\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{c}}_1+2\mathrm{AiryAi}(1,\mathit{\_}\mathit{Z}){\mathit{\_}\mathit{c}}_1)+{\mathit{\_}\mathit{h}}^2}d\mathit{\_}\mathit{h}+\mathit{\_}\mathit{C}\mathit{2})\right)}^{-1}d\mathit{\_}\mathit{f}+x+\mathit{\_}\mathit{C}\mathit{3})\frac{1}{\sqrt[3]{-3{\mathit{\_}\mathit{c}}_{1}t+\mathit{\_}\mathit{C}\mathit{4}}}$$ has RootOf

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