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'));
 

\[u \left (x , y , t\right ) = c_{4} +c_{5} \left (c_1 x +c_2 y +c_3 t +c_{4} \right )\]

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