4.19.41 $3x^4{y}^{\prime }(x{)}^2-xy(x)-y(x)=0$

ODE
$$3x^4{y}^{\prime }(x{)}^2-xy(x)-y(x)=0$$ 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

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

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

$$\{-\frac{\sqrt{3}}{3x}\sqrt{xy\left(x\right)+y\left(x\right)}\frac{1}{\sqrt{y\left(x\right)}}-\frac{\sqrt{3}}{3}\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(1\sqrt{xy\left(x\right)+y\left(x\right)}\frac{1}{\sqrt{y\left(x\right)}}\right)+2\sqrt{y\left(x\right)}+\mathit{\_}\mathit{C}\mathit{1}=0,\frac{\sqrt{3}}{3x}\sqrt{xy\left(x\right)+y\left(x\right)}\frac{1}{\sqrt{y\left(x\right)}}+\frac{\sqrt{3}}{3}\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(1\sqrt{xy\left(x\right)+y\left(x\right)}\frac{1}{\sqrt{y\left(x\right)}}\right)+2\sqrt{y\left(x\right)}+\mathit{\_}\mathit{C}\mathit{1}=0,y\left(x\right)=0\}$$ 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