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


\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


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

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