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 = 0.415011 (sec), leaf count = 120
\[\left \{\text {Solve}\left [6 c_1+\frac {2 \sqrt {x^2-4 x^8 y(x)} \tan ^{-1}\left (\sqrt {4 x^6 y(x)-1}\right )}{x \sqrt {4 x^6 y(x)-1}}+\log (y(x))=0,y(x)\right ],\text {Solve}\left [6 c_1+\log (y(x))=\frac {2 \sqrt {x^2-4 x^8 y(x)} \tan ^{-1}\left (\sqrt {4 x^6 y(x)-1}\right )}{x \sqrt {4 x^6 y(x)-1}},y(x)\right ]\right \}\]
Maple ✓
cpu = 0.069 (sec), leaf count = 71
\[ \left \{ \ln \left ( x \right ) -{\it \_C1}-{\frac {\ln \left ( y \left ( x \right ) {x}^{6} \right ) }{6}}-{\frac {1}{3}{\it Artanh} \left ( \sqrt {1-4\,y \left ( x \right ) {x}^{6}} \right ) }=0,\ln \left ( x \right ) -{\it \_C1}-{\frac {\ln \left ( y \left ( x \right ) {x}^{6} \right ) }{6}}+{\frac {1}{3}{\it Artanh} \left ( \sqrt {1-4\,y \left ( x \right ) {x}^{6}} \right ) }=0,y \left ( x \right ) ={\frac {1}{4\,{x}^{6}}} \right \} \] 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*ArcTan[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),'implicit')
Maple raw output
y(x) = 1/4/x^6, ln(x)-_C1-1/6*ln(y(x)*x^6)+1/3*arctanh((1-4*y(x)*x^6)^(1/2)) = 0, ln(x)-_C1-1/6*ln(y(x)*x^6)-1/3*arctanh((1-4*y(x)*x^6)^(1/2)) = 0