##### 4.17.29 $$-2 x^3 y(x)^2 y'(x)-4 x^2 y(x)^3+y'(x)^2=0$$

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

[[_1st_order, _with_linear_symmetries]]

Book solution method
Change of variable

Mathematica
cpu = 0.930627 (sec), leaf count = 136

$\left \{\text {Solve}\left [\frac {2 x \sqrt {x^4 y(x)+4} y(x)^{3/2} \sinh ^{-1}\left (\frac {1}{2} x^2 \sqrt {y(x)}\right )}{\sqrt {x^2 y(x)^3 \left (x^4 y(x)+4\right )}}+\log (y(x))+4 c_1=0,y(x)\right ],\text {Solve}\left [\log (y(x))+4 c_1=\frac {2 x y(x)^{3/2} \sqrt {x^4 y(x)+4} \sinh ^{-1}\left (\frac {1}{2} x^2 \sqrt {y(x)}\right )}{\sqrt {x^2 y(x)^3 \left (x^4 y(x)+4\right )}},y(x)\right ]\right \}$

Maple
cpu = 1.048 (sec), leaf count = 131

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

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

Mathematica raw output

{Solve[4*C[1] + Log[y[x]] + (2*x*ArcSinh[(x^2*Sqrt[y[x]])/2]*y[x]^(3/2)*Sqrt[4 +
 x^4*y[x]])/Sqrt[x^2*y[x]^3*(4 + x^4*y[x])] == 0, y[x]], Solve[4*C[1] + Log[y[x]
] == (2*x*ArcSinh[(x^2*Sqrt[y[x]])/2]*y[x]^(3/2)*Sqrt[4 + x^4*y[x]])/Sqrt[x^2*y[
x]^3*(4 + x^4*y[x])], y[x]]}

Maple raw input

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

Maple raw output

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