4.12.2 \(x (x-2 y(x)) y'(x)+(2 x-y(x)) y(x)=0\)

ODE
\[ x (x-2 y(x)) y'(x)+(2 x-y(x)) y(x)=0 \] ODE Classification

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0228071 (sec), leaf count = 62

\[\left \{\left \{y(x)\to \frac {1}{2} \left (x-\frac {\sqrt {x^3-4 e^{c_1}}}{\sqrt {x}}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (\frac {\sqrt {x^3-4 e^{c_1}}}{\sqrt {x}}+x\right )\right \}\right \}\]

Maple
cpu = 0.013 (sec), leaf count = 27

\[ \left \{ -{\frac {1}{3}\ln \left (-{\frac { \left (x-y \relax (x ) \right ) y \relax (x ) }{{x}^{2}}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[(2*x - y[x])*y[x] + x*(x - 2*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (x - Sqrt[-4*E^C[1] + x^3]/Sqrt[x])/2}, {y[x] -> (x + Sqrt[-4*E^C[1] +
 x^3]/Sqrt[x])/2}}

Maple raw input

dsolve(x*(x-2*y(x))*diff(y(x),x)+(2*x-y(x))*y(x) = 0, y(x),'implicit')

Maple raw output

-1/3*ln(-y(x)*(x-y(x))/x^2)-ln(x)-_C1 = 0