##### 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 = 0.247724 (sec), leaf count = 166

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

Maple
cpu = 1.979 (sec), leaf count = 209

$\left [y \left (x \right ) = 0, y \left (x \right ) = \frac {-2 \arctanh \left (\sqrt {x +1}\right ) \textit {\_C1} \,x^{2} \sqrt {3}-\frac {2 \sqrt {3}\, \textit {\_C1} \,x^{2}}{\sqrt {x +1}}+\arctanh \left (\sqrt {x +1}\right )^{2} x^{2}+\frac {2 \arctanh \left (\sqrt {x +1}\right ) x^{2}}{\sqrt {x +1}}-\frac {2 \sqrt {3}\, \textit {\_C1} x}{\sqrt {x +1}}+3 \textit {\_C1}^{2} x^{2}+\frac {2 \arctanh \left (\sqrt {x +1}\right ) x}{\sqrt {x +1}}+x +1}{12 x^{2}}, y \left (x \right ) = \frac {2 \arctanh \left (\sqrt {x +1}\right ) \textit {\_C1} \,x^{2} \sqrt {3}+\frac {2 \sqrt {3}\, \textit {\_C1} \,x^{2}}{\sqrt {x +1}}+\arctanh \left (\sqrt {x +1}\right )^{2} x^{2}+\frac {2 \arctanh \left (\sqrt {x +1}\right ) x^{2}}{\sqrt {x +1}}+\frac {2 \sqrt {3}\, \textit {\_C1} x}{\sqrt {x +1}}+3 \textit {\_C1}^{2} x^{2}+\frac {2 \arctanh \left (\sqrt {x +1}\right ) x}{\sqrt {x +1}}+x +1}{12 x^{2}}\right ]$ 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))

Maple raw output

[y(x) = 0, y(x) = 1/12*(-2*arctanh((x+1)^(1/2))*_C1*x^2*3^(1/2)-2*3^(1/2)/(x+1)^
(1/2)*_C1*x^2+arctanh((x+1)^(1/2))^2*x^2+2*arctanh((x+1)^(1/2))/(x+1)^(1/2)*x^2-
2*3^(1/2)/(x+1)^(1/2)*_C1*x+3*_C1^2*x^2+2*arctanh((x+1)^(1/2))/(x+1)^(1/2)*x+x+1
)/x^2, y(x) = 1/12*(2*arctanh((x+1)^(1/2))*_C1*x^2*3^(1/2)+2*3^(1/2)/(x+1)^(1/2)
*_C1*x^2+arctanh((x+1)^(1/2))^2*x^2+2*arctanh((x+1)^(1/2))/(x+1)^(1/2)*x^2+2*3^(
1/2)/(x+1)^(1/2)*_C1*x+3*_C1^2*x^2+2*arctanh((x+1)^(1/2))/(x+1)^(1/2)*x+x+1)/x^2
]