ODE
\[ x^6 y'(x)^2-2 x y'(x)-4 y(x)=0 \] ODE Classification
[[_homogeneous, `class G`], _rational]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 0.385876 (sec), leaf count = 120
\[\left \{\text {Solve}\left [4 c_1+\frac {2 \sqrt {4 x^6 y(x)+x^2} \tanh ^{-1}\left (\sqrt {4 x^4 y(x)+1}\right )}{x \sqrt {4 x^4 y(x)+1}}+\log (y(x))=0,y(x)\right ],\text {Solve}\left [4 c_1+\log (y(x))=\frac {2 \sqrt {4 x^6 y(x)+x^2} \tanh ^{-1}\left (\sqrt {4 x^4 y(x)+1}\right )}{x \sqrt {4 x^4 y(x)+1}},y(x)\right ]\right \}\]
Maple ✓
cpu = 0.063 (sec), leaf count = 71
\[ \left \{ \ln \left ( x \right ) -{\it \_C1}-{\frac {\ln \left ( {x}^{4}y \left ( x \right ) \right ) }{4}}-{\frac {1}{2}{\it Artanh} \left ( \sqrt {4\,{x}^{4}y \left ( x \right ) +1} \right ) }=0,\ln \left ( x \right ) -{\it \_C1}-{\frac {\ln \left ( {x}^{4}y \left ( x \right ) \right ) }{4}}+{\frac {1}{2}{\it Artanh} \left ( \sqrt {4\,{x}^{4}y \left ( x \right ) +1} \right ) }=0,y \left ( x \right ) =-{\frac {1}{4\,{x}^{4}}} \right \} \] Mathematica raw input
DSolve[-4*y[x] - 2*x*y'[x] + x^6*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{Solve[4*C[1] + Log[y[x]] + (2*ArcTanh[Sqrt[1 + 4*x^4*y[x]]]*Sqrt[x^2 + 4*x^6*y[x]])/(x*Sqrt[1 + 4*x^4*y[x]]) == 0, y[x]], Solve[4*C[1] + Log[y[x]] == (2*ArcTanh[Sqrt[1 + 4*x^4*y[x]]]*Sqrt[x^2 + 4*x^6*y[x]])/(x*Sqrt[1 + 4*x^4*y[x]]), y[x]]}
Maple raw input
dsolve(x^6*diff(y(x),x)^2-2*x*diff(y(x),x)-4*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = -1/4/x^4, ln(x)-_C1-1/4*ln(x^4*y(x))-1/2*arctanh((4*x^4*y(x)+1)^(1/2)) = 0, ln(x)-_C1-1/4*ln(x^4*y(x))+1/2*arctanh((4*x^4*y(x)+1)^(1/2)) = 0