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'));
 

\[u \left (x , t\right ) = \frac {\operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {1}{\operatorname {RootOf}\left (-\ln \left (\textit {\_f} \right )+2 \int _{}^{\textit {\_Z}}\frac {\textit {\_h}}{2 \,2^{{1}/{3}} \left (-\textit {\_c}_{1}^{2}\right )^{{1}/{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\textit {\_Z} \right ) 2^{{1}/{3}} \left (-\textit {\_c}_{1}^{2}\right )^{{1}/{3}} c_{1} \textit {\_h} +2^{{1}/{3}} \left (-\textit {\_c}_{1}^{2}\right )^{{1}/{3}} \textit {\_h} \operatorname {AiryAi}\left (\textit {\_Z} \right )+2 \operatorname {AiryBi}\left (1, \textit {\_Z}\right ) c_{1} \textit {\_c}_{1}+2 \operatorname {AiryAi}\left (1, \textit {\_Z}\right ) \textit {\_c}_{1}\right )+\textit {\_h}^{2}}d \textit {\_h} +c_{2} \right )}d \textit {\_f} +x +c_{3} \right )}{\left (-3 \textit {\_c}_{1} t +c_{4} \right )^{{1}/{3}}}\]

has RootOf

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