##### 4.19.44 $$x^8 y'(x)^2+3 x y'(x)+9 y(x)=0$$

ODE
$x^8 y'(x)^2+3 x y'(x)+9 y(x)=0$ ODE Classiﬁcation

[[_homogeneous, class G]]

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

Mathematica
cpu = 0.628343 (sec), leaf count = 120

$\left \{\text {Solve}\left [\frac {2 \sqrt {x^2-4 x^8 y(x)} \tan ^{-1}\left (\sqrt {4 x^6 y(x)-1}\right )}{x \sqrt {4 x^6 y(x)-1}}+\log (y(x))+6 c_1=0,y(x)\right ],\text {Solve}\left [\log (y(x))+6 c_1=\frac {2 \sqrt {x^2-4 x^8 y(x)} \tan ^{-1}\left (\sqrt {4 x^6 y(x)-1}\right )}{x \sqrt {4 x^6 y(x)-1}},y(x)\right ]\right \}$

Maple
cpu = 8.829 (sec), leaf count = 42

$\left [y \left (x \right ) = \frac {1}{4 x^{6}}, y \left (x \right ) = \frac {-x^{3}+\textit {\_C1}}{x^{3} \textit {\_C1}^{2}}, y \left (x \right ) = -\frac {x^{3}+\textit {\_C1}}{x^{3} \textit {\_C1}^{2}}\right ]$ Mathematica raw input

DSolve[9*y[x] + 3*x*y'[x] + x^8*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[6*C[1] + Log[y[x]] + (2*ArcTan[Sqrt[-1 + 4*x^6*y[x]]]*Sqrt[x^2 - 4*x^8*y[
x]])/(x*Sqrt[-1 + 4*x^6*y[x]]) == 0, y[x]], Solve[6*C[1] + Log[y[x]] == (2*ArcTa
n[Sqrt[-1 + 4*x^6*y[x]]]*Sqrt[x^2 - 4*x^8*y[x]])/(x*Sqrt[-1 + 4*x^6*y[x]]), y[x]
]}

Maple raw input

dsolve(x^8*diff(y(x),x)^2+3*x*diff(y(x),x)+9*y(x) = 0, y(x))

Maple raw output

[y(x) = 1/4/x^6, y(x) = (-x^3+_C1)/x^3/_C1^2, y(x) = -1/x^3*(x^3+_C1)/_C1^2]