##### 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 = 0.298767 (sec), leaf count = 62

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

Maple
cpu = 0.181 (sec), leaf count = 69

$\left [y \left (x \right ) = \frac {\textit {\_C1}^{2} x^{2}-\sqrt {x^{4} \textit {\_C1}^{4}+4 x \textit {\_C1}}}{2 x \,\textit {\_C1}^{2}}, y \left (x \right ) = \frac {\textit {\_C1}^{2} x^{2}+\sqrt {x^{4} \textit {\_C1}^{4}+4 x \textit {\_C1}}}{2 x \,\textit {\_C1}^{2}}\right ]$ 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))

Maple raw output

[y(x) = 1/2/x/_C1^2*(_C1^2*x^2-(_C1^4*x^4+4*_C1*x)^(1/2)), y(x) = 1/2/x/_C1^2*(_
C1^2*x^2+(_C1^4*x^4+4*_C1*x)^(1/2))]