4.19.41 \(3 x^4 y'(x)^2-x y(x)-y(x)=0\)

ODE
\[ 3 x^4 y'(x)^2-x y(x)-y(x)=0 \] ODE Classification

[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica
cpu = 0.249133 (sec), leaf count = 166

\[\left \{\left \{y(x)\to \frac {3 c_1^2 x^2-2 \sqrt {3} c_1 x \sqrt {x+1}+2 x \left (\sqrt {x+1}-\sqrt {3} c_1 x\right ) \tanh ^{-1}\left (\sqrt {x+1}\right )+x^2 \tanh ^{-1}\left (\sqrt {x+1}\right )^2+x+1}{12 x^2}\right \},\left \{y(x)\to \frac {3 c_1^2 x^2+2 \sqrt {3} c_1 x \sqrt {x+1}+2 x \left (\sqrt {3} c_1 x+\sqrt {x+1}\right ) \tanh ^{-1}\left (\sqrt {x+1}\right )+x^2 \tanh ^{-1}\left (\sqrt {x+1}\right )^2+x+1}{12 x^2}\right \}\right \}\]

Maple
cpu = 0.098 (sec), leaf count = 107

\[ \left \{ -{\frac {\sqrt {3}}{3\,x}\sqrt {xy \relax (x ) +y \relax (x ) }{\frac {1}{\sqrt {y \relax (x ) }}}}-{\frac {\sqrt {3}}{3}{\it Artanh} \left ({1\sqrt {xy \relax (x ) +y \relax (x ) }{\frac {1}{\sqrt {y \relax (x ) }}}} \right ) }+2\,\sqrt {y \relax (x ) }+{\it \_C1}=0,{\frac {\sqrt {3}}{3\,x}\sqrt {xy \relax (x ) +y \relax (x ) }{\frac {1}{\sqrt {y \relax (x ) }}}}+{\frac {\sqrt {3}}{3}{\it Artanh} \left ({1\sqrt {xy \relax (x ) +y \relax (x ) }{\frac {1}{\sqrt {y \relax (x ) }}}} \right ) }+2\,\sqrt {y \relax (x ) }+{\it \_C1}=0,y \relax (x ) =0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (1 + x + x^2*ArcTanh[Sqrt[1 + x]]^2 - 2*Sqrt[3]*x*Sqrt[1 + x]*C[1] + 3
*x^2*C[1]^2 + 2*x*ArcTanh[Sqrt[1 + x]]*(Sqrt[1 + x] - Sqrt[3]*x*C[1]))/(12*x^2)}
, {y[x] -> (1 + x + x^2*ArcTanh[Sqrt[1 + x]]^2 + 2*Sqrt[3]*x*Sqrt[1 + x]*C[1] + 
3*x^2*C[1]^2 + 2*x*ArcTanh[Sqrt[1 + x]]*(Sqrt[1 + x] + Sqrt[3]*x*C[1]))/(12*x^2)
}}

Maple raw input

dsolve(3*x^4*diff(y(x),x)^2-x*y(x)-y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = 0, -1/3*3^(1/2)/y(x)^(1/2)*(x*y(x)+y(x))^(1/2)/x-1/3*3^(1/2)*arctanh((x*y
(x)+y(x))^(1/2)/y(x)^(1/2))+2*y(x)^(1/2)+_C1 = 0, 1/3*3^(1/2)/y(x)^(1/2)*(x*y(x)
+y(x))^(1/2)/x+1/3*3^(1/2)*arctanh((x*y(x)+y(x))^(1/2)/y(x)^(1/2))+2*y(x)^(1/2)+
_C1 = 0