4.11.13 \(x y(x) y'(x)=x^2-x y(x)+y(x)^2\)

ODE
\[ x y(x) y'(x)=x^2-x y(x)+y(x)^2 \] ODE Classification

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type``class B`]]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0185526 (sec), leaf count = 20

\[\left \{\left \{y(x)\to x \left (W\left (\frac {e^{c_1-1}}{x}\right )+1\right )\right \}\right \}\]

Maple
cpu = 0.01 (sec), leaf count = 26

\[ \left \{ -{\it \_C1}+\ln \left ({\frac {y \relax (x ) -x}{x}} \right ) +{\frac {y \relax (x ) }{x}}+\ln \relax (x ) =0 \right \} \] Mathematica raw input

DSolve[x*y[x]*y'[x] == x^2 - x*y[x] + y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> x*(1 + ProductLog[E^(-1 + C[1])/x])}}

Maple raw input

dsolve(x*y(x)*diff(y(x),x) = x^2-x*y(x)+y(x)^2, y(x),'implicit')

Maple raw output

-_C1+ln((y(x)-x)/x)+y(x)/x+ln(x) = 0