##### 4.19.8 $$x \left (x^3-2 y(x)\right ) y'(x)-\left (2 x^3-y(x)\right ) y(x)+x^2 y'(x)^2=0$$

ODE
$x \left (x^3-2 y(x)\right ) y'(x)-\left (2 x^3-y(x)\right ) y(x)+x^2 y'(x)^2=0$ ODE Classiﬁcation

[[_homogeneous, class G], _rational]

Book solution method
Change of variable

Mathematica
cpu = 1.9072 (sec), leaf count = 154

$\left \{\text {Solve}\left [\frac {1}{2} \left (-\frac {2 x^{5/2} \sqrt {\frac {x^3}{y(x)}+4} \sqrt {y(x)} \sinh ^{-1}\left (\frac {x^{3/2}}{2 \sqrt {y(x)}}\right )}{\sqrt {x^8+4 x^5 y(x)}}+\log (y(x))-\log (x)\right )=c_1,y(x)\right ],\text {Solve}\left [\frac {x^{5/2} \sqrt {\frac {x^3}{y(x)}+4} \sqrt {y(x)} \sinh ^{-1}\left (\frac {x^{3/2}}{2 \sqrt {y(x)}}\right )}{\sqrt {x^8+4 x^5 y(x)}}+\frac {1}{2} \log (y(x))-\frac {\log (x)}{2}=c_1,y(x)\right ]\right \}$

Maple
cpu = 0.595 (sec), leaf count = 115

$\left [y \left (x \right ) = -\frac {x^{3}}{4}, y \left (x \right ) = -\frac {x^{3}}{2}+x \,\textit {\_C1}^{2}-\frac {\left (-x -2 \textit {\_C1} \right ) x^{2}}{2}, y \left (x \right ) = -\frac {x^{3}}{2}+x \,\textit {\_C1}^{2}-\frac {\left (-x +2 \textit {\_C1} \right ) x^{2}}{2}, y \left (x \right ) = \frac {x \left (\left (x \textit {\_C1} -2\right ) \textit {\_C1} x -\textit {\_C1}^{2} x^{2}+2\right )}{2 \textit {\_C1}^{2}}, y \left (x \right ) = \frac {x \left (\left (x \textit {\_C1} +2\right ) \textit {\_C1} x -\textit {\_C1}^{2} x^{2}+2\right )}{2 \textit {\_C1}^{2}}\right ]$ Mathematica raw input

DSolve[-((2*x^3 - y[x])*y[x]) + x*(x^3 - 2*y[x])*y'[x] + x^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[(-Log[x] + Log[y[x]] - (2*x^(5/2)*ArcSinh[x^(3/2)/(2*Sqrt[y[x]])]*Sqrt[4
+ x^3/y[x]]*Sqrt[y[x]])/Sqrt[x^8 + 4*x^5*y[x]])/2 == C[1], y[x]], Solve[-1/2*Log
[x] + Log[y[x]]/2 + (x^(5/2)*ArcSinh[x^(3/2)/(2*Sqrt[y[x]])]*Sqrt[4 + x^3/y[x]]*
Sqrt[y[x]])/Sqrt[x^8 + 4*x^5*y[x]] == C[1], y[x]]}

Maple raw input

dsolve(x^2*diff(y(x),x)^2+x*(x^3-2*y(x))*diff(y(x),x)-(2*x^3-y(x))*y(x) = 0, y(x))

Maple raw output

[y(x) = -1/4*x^3, y(x) = -1/2*x^3+x*_C1^2-1/2*(-x-2*_C1)*x^2, y(x) = -1/2*x^3+x*
_C1^2-1/2*(-x+2*_C1)*x^2, y(x) = 1/2*x*((_C1*x-2)*_C1*x-_C1^2*x^2+2)/_C1^2, y(x)
 = 1/2*x*((_C1*x+2)*_C1*x-_C1^2*x^2+2)/_C1^2]