##### 4.17.46 $$3 x y(x)^4 y'(x)+9 y'(x)^2+y(x)^5=0$$

ODE
$3 x y(x)^4 y'(x)+9 y'(x)^2+y(x)^5=0$ ODE Classiﬁcation

[[_1st_order, _with_linear_symmetries]]

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

Mathematica
cpu = 1.12193 (sec), leaf count = 156

$\left \{\text {Solve}\left [\frac {\sqrt {x^2 y(x)^3-4} y(x)^{5/2} \tanh ^{-1}\left (\frac {x y(x)^{3/2}}{\sqrt {x^2 y(x)^3-4}}\right )}{\sqrt {y(x)^5 \left (x^2 y(x)^3-4\right )}}+\frac {3}{2} \log (y(x))+c_1=0,y(x)\right ],\text {Solve}\left [\frac {y(x)^{5/2} \sqrt {x^2 y(x)^3-4} \tanh ^{-1}\left (\frac {x y(x)^{3/2}}{\sqrt {x^2 y(x)^3-4}}\right )}{\sqrt {y(x)^5 \left (x^2 y(x)^3-4\right )}}=\frac {3}{2} \log (y(x))+c_1,y(x)\right ]\right \}$

Maple
cpu = 1.936 (sec), leaf count = 105

$\left [y \left (x \right ) = \frac {4^{\frac {1}{3}}}{x^{\frac {2}{3}}}, y \left (x \right ) = -\frac {4^{\frac {1}{3}}}{2 x^{\frac {2}{3}}}-\frac {i \sqrt {3}\, 4^{\frac {1}{3}}}{2 x^{\frac {2}{3}}}, y \left (x \right ) = -\frac {4^{\frac {1}{3}}}{2 x^{\frac {2}{3}}}+\frac {i \sqrt {3}\, 4^{\frac {1}{3}}}{2 x^{\frac {2}{3}}}, y \left (x \right ) = \frac {\RootOf \left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\frac {3 \textit {\_a}^{3}}{2}+\frac {3 \sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4\right )}}{2}-6}{\textit {\_a} \left (\textit {\_a}^{3}-4\right )}d \textit {\_a} +\textit {\_C1} \right )}{x^{\frac {2}{3}}}\right ]$ Mathematica raw input

DSolve[y[x]^5 + 3*x*y[x]^4*y'[x] + 9*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[C[1] + (3*Log[y[x]])/2 + (ArcTanh[(x*y[x]^(3/2))/Sqrt[-4 + x^2*y[x]^3]]*y
[x]^(5/2)*Sqrt[-4 + x^2*y[x]^3])/Sqrt[y[x]^5*(-4 + x^2*y[x]^3)] == 0, y[x]], Sol
ve[(ArcTanh[(x*y[x]^(3/2))/Sqrt[-4 + x^2*y[x]^3]]*y[x]^(5/2)*Sqrt[-4 + x^2*y[x]^
3])/Sqrt[y[x]^5*(-4 + x^2*y[x]^3)] == C[1] + (3*Log[y[x]])/2, y[x]]}

Maple raw input

dsolve(9*diff(y(x),x)^2+3*x*y(x)^4*diff(y(x),x)+y(x)^5 = 0, y(x))

Maple raw output

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