2.1.22 Clairaut equation. \(u = x u_x+ y u_y + \sin ( u_x + u_y )\)

problem number 22

Taken from Mathematica DSolve help pages

Another example of nonlinear Clairaut equation

Solve for \(u(x,y)\)

\[ u = x u_x+ y u_y + \sin ( u_x + u_y ) \]

Mathematica ✓

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

\[\{\{u(x,y)\to c_1 x+c_2 y+\sin (c_1+c_2)\}\}\]

Maple ✓

restart; 
pde := u(x,y)= x*diff(u(x,y),x)+y*diff(u(x,y),y)+sin(diff(u(x,y),x)+diff(u(x,y),y)); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y))),output='realtime'));
 

\[u \left (x , y\right ) = x \textit {\_c}_{1}+y \textit {\_c}_{2}+\sin \left (\textit {\_c}_{1}+\textit {\_c}_{2}\right )\]

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