##### 4.19.43 $$x^6 y'(x)^2-2 x y'(x)-4 y(x)=0$$

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

[[_homogeneous, class G], _rational]

Book solution method
No Missing Variables ODE, Solve for $$y$$

Mathematica
cpu = 0.612382 (sec), leaf count = 120

$\left \{\text {Solve}\left [\frac {2 \sqrt {4 x^6 y(x)+x^2} \tanh ^{-1}\left (\sqrt {4 x^4 y(x)+1}\right )}{x \sqrt {4 x^4 y(x)+1}}+\log (y(x))+4 c_1=0,y(x)\right ],\text {Solve}\left [\log (y(x))+4 c_1=\frac {2 \sqrt {4 x^6 y(x)+x^2} \tanh ^{-1}\left (\sqrt {4 x^4 y(x)+1}\right )}{x \sqrt {4 x^4 y(x)+1}},y(x)\right ]\right \}$

Maple
cpu = 8.171 (sec), leaf count = 143

$\left [y \left (x \right ) = -\frac {1}{4 x^{4}}, y \left (x \right ) = \frac {-2 x^{4}-\textit {\_C1}^{2}-\textit {\_C1} \left (2 i x^{2}-\textit {\_C1} \right )}{2 \textit {\_C1}^{2} x^{4}}, y \left (x \right ) = \frac {-2 x^{4}-\textit {\_C1}^{2}-\textit {\_C1} \left (-2 i x^{2}-\textit {\_C1} \right )}{2 \textit {\_C1}^{2} x^{4}}, y \left (x \right ) = \frac {-2 x^{4}+\textit {\_C1} \left (2 i x^{2}+\textit {\_C1} \right )-\textit {\_C1}^{2}}{2 \textit {\_C1}^{2} x^{4}}, y \left (x \right ) = \frac {-2 x^{4}+\textit {\_C1} \left (-2 i x^{2}+\textit {\_C1} \right )-\textit {\_C1}^{2}}{2 \textit {\_C1}^{2} x^{4}}\right ]$ Mathematica raw input

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

Mathematica raw output

{Solve[4*C[1] + Log[y[x]] + (2*ArcTanh[Sqrt[1 + 4*x^4*y[x]]]*Sqrt[x^2 + 4*x^6*y[
x]])/(x*Sqrt[1 + 4*x^4*y[x]]) == 0, y[x]], Solve[4*C[1] + Log[y[x]] == (2*ArcTan
h[Sqrt[1 + 4*x^4*y[x]]]*Sqrt[x^2 + 4*x^6*y[x]])/(x*Sqrt[1 + 4*x^4*y[x]]), y[x]]}

Maple raw input

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

Maple raw output

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