2.1.42 \(y^2 u_x- x y u_y=x(u-2 y)\) Problem 3.3(g) Lokenath Debnath

problem number 42

Added June 3, 2019.

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

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

Mathematica

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

\begin {align*} & \left \{u(x,y)\to \frac {\sqrt {y^2} c_1\left (\frac {1}{2} \left (x^2+y^2\right )\right )-x^2 \sqrt {-y^2}}{\sqrt {-y^4}}\right \}\\& \left \{u(x,y)\to \frac {\sqrt {y^2} c_1\left (\frac {1}{2} \left (x^2+y^2\right )\right )+x^2 \sqrt {-y^2}}{\sqrt {-y^4}}\right \}\\ \end {align*}

Maple

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

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

____________________________________________________________________________________