##### 4.18.44 $$8 y(x) y'(x)+16 x y'(x)^2+y(x)^6=0$$

ODE
$8 y(x) y'(x)+16 x y'(x)^2+y(x)^6=0$ ODE Classiﬁcation

[[_homogeneous, class G]]

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

Mathematica
cpu = 0.542397 (sec), leaf count = 58

$\left \{\left \{y(x)\to -\frac {\sqrt {2} e^{\frac {c_1}{4}}}{\sqrt {x+e^{c_1}}}\right \},\left \{y(x)\to \frac {\sqrt {2} e^{\frac {c_1}{4}}}{\sqrt {x+e^{c_1}}}\right \}\right \}$

Maple
cpu = 0.239 (sec), leaf count = 99

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

DSolve[y[x]^6 + 8*y[x]*y'[x] + 16*x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((Sqrt[2]*E^(C[1]/4))/Sqrt[E^C[1] + x])}, {y[x] -> (Sqrt[2]*E^(C[1]/4
))/Sqrt[E^C[1] + x]}}

Maple raw input

dsolve(16*x*diff(y(x),x)^2+8*y(x)*diff(y(x),x)+y(x)^6 = 0, y(x))

Maple raw output

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