##### 4.20.27 $$-6 x^3 y'(x)+4 x^2 y(x)+y(x)^2 y'(x)^2=0$$

ODE
$-6 x^3 y'(x)+4 x^2 y(x)+y(x)^2 y'(x)^2=0$ ODE Classiﬁcation

[[_1st_order, _with_linear_symmetries]]

Book solution method
Change of variable

Mathematica
cpu = 0.63946 (sec), leaf count = 153

$\left \{\text {Solve}\left [\frac {2 \sqrt {9 x^6-4 x^2 y(x)^3} \tanh ^{-1}\left (\frac {3 x^2}{\sqrt {9 x^4-4 y(x)^3}}\right )}{x \sqrt {9 x^4-4 y(x)^3}}+4 c_1=3 \log (y(x)),y(x)\right ],\text {Solve}\left [4 c_1=\frac {2 \sqrt {9 x^6-4 x^2 y(x)^3} \tanh ^{-1}\left (\frac {3 x^2}{\sqrt {9 x^4-4 y(x)^3}}\right )}{x \sqrt {9 x^4-4 y(x)^3}}+3 \log (y(x)),y(x)\right ]\right \}$

Maple
cpu = 1.348 (sec), leaf count = 114

$\left [y \left (x \right ) = \frac {18^{\frac {1}{3}} x^{\frac {4}{3}}}{2}, y \left (x \right ) = \left (-\frac {18^{\frac {1}{3}} x^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 18^{\frac {1}{3}} x^{\frac {1}{3}}}{4}\right ) x, y \left (x \right ) = \left (-\frac {18^{\frac {1}{3}} x^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 18^{\frac {1}{3}} x^{\frac {1}{3}}}{4}\right ) x, y \left (x \right ) = \RootOf \left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {-3 \textit {\_a}^{3}+\frac {9 \sqrt {-4 \textit {\_a}^{3}+9}}{4}+\frac {27}{4}}{\textit {\_a} \left (4 \textit {\_a}^{3}-9\right )}d \textit {\_a} +\textit {\_C1} \right ) x^{\frac {4}{3}}\right ]$ Mathematica raw input

DSolve[4*x^2*y[x] - 6*x^3*y'[x] + y[x]^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[4*C[1] + (2*ArcTanh[(3*x^2)/Sqrt[9*x^4 - 4*y[x]^3]]*Sqrt[9*x^6 - 4*x^2*y[
x]^3])/(x*Sqrt[9*x^4 - 4*y[x]^3]) == 3*Log[y[x]], y[x]], Solve[4*C[1] == 3*Log[y
[x]] + (2*ArcTanh[(3*x^2)/Sqrt[9*x^4 - 4*y[x]^3]]*Sqrt[9*x^6 - 4*x^2*y[x]^3])/(x
*Sqrt[9*x^4 - 4*y[x]^3]), y[x]]}

Maple raw input

dsolve(y(x)^2*diff(y(x),x)^2-6*x^3*diff(y(x),x)+4*x^2*y(x) = 0, y(x))

Maple raw output

[y(x) = 1/2*18^(1/3)*x^(4/3), y(x) = (-1/4*18^(1/3)*x^(1/3)-1/4*I*3^(1/2)*18^(1/
3)*x^(1/3))*x, y(x) = (-1/4*18^(1/3)*x^(1/3)+1/4*I*3^(1/2)*18^(1/3)*x^(1/3))*x,
y(x) = RootOf(-ln(x)+Intat(3/4*(-4*_a^3+3*(-4*_a^3+9)^(1/2)+9)/_a/(4*_a^3-9),_a
= _Z)+_C1)*x^(4/3)]