2.1.39 \(2 x y u_x+(x^2+y^2)u_y=0\) Problem 3.3(e) Lokenath Debnath

problem number 39

Added June 2, 2019.

Problem 3.3(e) nonlinear pde’s by Lokenath Debnath, 3rd edition.

Solve for \(u(x,y)\) \[ 2 x y u_x+(x^2+y^2)u_y=0 \]

Mathematica

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

\[\left \{\left \{u(x,y)\to c_1\left (\frac {y^2}{x}-x\right )\right \}\right \}\]

Maple

restart; 
pde :=2*x*y*diff(u(x,y),x)+(x^2+y^2)*diff(u(x,y),y)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y))),output='realtime'));
 

\[u \left (x , y\right ) = \mathit {\_F1} \left (\frac {-x^{2}+y^{2}}{x}\right )\]

____________________________________________________________________________________