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

problem number 41

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 \fbox {$\frac {\sqrt {y^2} c_1\left (\frac {1}{2} \left (x^2+y^2\right )\right )-x^2 \sqrt {-y^2}}{\sqrt {-y^4}}\text { if }x=0\lor y\geq 0$}\right \}\\& \left \{u(x,y)\to \fbox {$\frac {\sqrt {-y^2} x^2+\sqrt {y^2} c_1\left (\frac {1}{2} \left (x^2+y^2\right )\right )}{\sqrt {-y^4}}\text { if }x=0\lor y\leq 0$}\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'));
 

\[u \left (x , y\right ) = -\frac {x^{2}}{y}+\frac {f_{1} \left (x^{2}+y^{2}\right )}{\sqrt {-y^{2}}}\]

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