2.1.68 \(x u(u^2+x y)u_x - y u(u^2+x y) u_y = x^4\). Problem 3.17(b) Lokenath Debnath

problem number 68

Added June 3, 2019.

Problem 3.17(b) nonlinear pde’s by Lokenath Debnath, 3rd edition.

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

\[ x u(u^2+x y)u_x - y u(u^2+x y) u_y = x^4 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*u[x,y]*(u[x,y]^2+x*y)*D[u[x, y], x] -y*u[x,y]*(u[x,y]^2+x*y)*D[u[x, y], y]== x^4; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde ,u[x, y], {x, y}], 60*10]];
 

\begin{align*}& \left \{u(x,y)\to -\sqrt {-x y-\sqrt {4 c_1(x y)+x^4+x^2 y^2}}\right \}\\& \left \{u(x,y)\to \sqrt {-x y-\sqrt {4 c_1(x y)+x^4+x^2 y^2}}\right \}\\& \left \{u(x,y)\to -\sqrt {-x y+\sqrt {4 c_1(x y)+x^4+x^2 y^2}}\right \}\\& \left \{u(x,y)\to \sqrt {-x y+\sqrt {4 c_1(x y)+x^4+x^2 y^2}}\right \}\\\end{align*}

Maple ✓

restart; 
pde :=x*u(x,y)*(u(x,y)^2+x*y)*diff(u(x,y),x) -y*u(x,y)*(u(x,y)^2+x*y)*diff(u(x,y),y)=x^4; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y)) ),output='realtime'));
 

\[u \left (x , y\right ) = \sqrt {-x y -\sqrt {x^{4}+4 f_{1} \left (x y \right )}}\]

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