4.18.42 $4y(x)y^{\prime }(x)+4x{y}^{\prime }(x{)}^2-y(x{)}^4=0$

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

[[_homogeneous, `class G`]]

Book solution method
No Missing Variables ODE, Solve for $x$

Mathematica ✓
cpu = 0.194342 (sec), leaf count = 123

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

Maple ✓
cpu = 0.073 (sec), leaf count = 53

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

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

Mathematica raw output

{{y[x] -> -(Sqrt[-1 + Tanh[(C[1] - Log[x])/2]^2]/Sqrt[x])}, {y[x] -> Sqrt[-1 + T
anh[(C[1] - Log[x])/2]^2]/Sqrt[x]}, {y[x] -> -(Sqrt[-1 + Tanh[(-C[1] + Log[x])/2
]^2]/Sqrt[x])}, {y[x] -> Sqrt[-1 + Tanh[(-C[1] + Log[x])/2]^2]/Sqrt[x]}}

Maple raw input

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

Maple raw output

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