##### 4.20.2 $$x^3 y'(x)-x^2 y(x)+y(x) y'(x)^2=0$$

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

[[_1st_order, _with_linear_symmetries]]

Book solution method
Change of variable

Mathematica
cpu = 0.594666 (sec), leaf count = 133

$\left \{\text {Solve}\left [\frac {\sqrt {x^6+4 x^2 y(x)^2} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+4 y(x)^2}}\right )}{x \sqrt {x^4+4 y(x)^2}}+2 c_1=\log (y(x)),y(x)\right ],\text {Solve}\left [2 c_1=\frac {\sqrt {x^6+4 x^2 y(x)^2} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+4 y(x)^2}}\right )}{x \sqrt {x^4+4 y(x)^2}}+\log (y(x)),y(x)\right ]\right \}$

Maple
cpu = 1.96 (sec), leaf count = 87

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

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

Mathematica raw output

{Solve[2*C[1] + (ArcTanh[x^2/Sqrt[x^4 + 4*y[x]^2]]*Sqrt[x^6 + 4*x^2*y[x]^2])/(x*
Sqrt[x^4 + 4*y[x]^2]) == Log[y[x]], y[x]], Solve[2*C[1] == Log[y[x]] + (ArcTanh[
x^2/Sqrt[x^4 + 4*y[x]^2]]*Sqrt[x^6 + 4*x^2*y[x]^2])/(x*Sqrt[x^4 + 4*y[x]^2]), y[
x]]}

Maple raw input

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

Maple raw output

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