2.15.21 Lin Tsien \(2 u_{tx} + u_x u_{xx} - u_{yy} = 0\)

problem number 130

Added December 27, 2018.

Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations

Lin Tsien equation. Solve for \(u(x,y,t)\) \[ 2 u_{tx} + u_x u_{xx} - u_{yy} = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  2*D[u[x, y, t], t, x] + D[u[x, y, t], x]*D[u[x, y, t], {x, 2}] - D[u[x, y, t], {y, 2}] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y, t], {x, y, t}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=2*diff(u(x,y,t),t,x)+ diff(u(x,y,t),x)* diff(u(x,y,t),x$2) - diff(u(x,y,t),y$2) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y,t))),output='realtime'));
 

\[u \left ( x,y,t \right ) ={\it \_C4}+{\it \_C5}\, \left ( {\it \_C1}\,x+{\it \_C2}\,y+{\it \_C3}\,t+{\it \_C4} \right ) \]

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