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 = 1.15263 (sec), leaf count = 193

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

Maple
cpu = 0.452 (sec), leaf count = 60

\[ \left \{ \left (y \relax (x ) \right ) ^{3}-4\,{x}^{-2}=0,\ln \relax (x ) -\int ^{y \relax (x ) {x}^{{\frac {2}{3}}}}\!{\frac {3}{2\,{\it \_a}\, \left ({{\it \_a}}^{3}-4 \right ) } \left ({{\it \_a}}^{3}+\sqrt {{{\it \_a}}^{3} \left ({{\it \_a}}^{3}-4 \right ) }-4 \right ) }{d{\it \_a}}-{\it \_C1}=0 \right \} \] 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[(-2*C[1] - 3*Log[y[x]] + ((3*Log[y[x]] - 2*Log[x*y[x]^3 + 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)])/2 == 0, y[x]], Solve[(-2*C[1] - 3*Log[y[x]] + ((-3*Log[y[x]] + 2*Log[x*y[x
]^3 + 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)])/2 == 0, 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),'implicit')

Maple raw output

y(x)^3-4/x^2 = 0, ln(x)-Intat(3/2*(_a^3+(_a^3*(_a^3-4))^(1/2)-4)/_a/(_a^3-4),_a 
= y(x)*x^(2/3))-_C1 = 0