4.18.13 $y(x)y^{\prime }(x)+x{y}^{\prime }(x{)}^2-y(x{)}^4=0$

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

[[_homogeneous, `class G`]]

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

Mathematica ✓
cpu = 0.204213 (sec), leaf count = 133

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

Maple ✓
cpu = 0.072 (sec), leaf count = 57

$$\{{\left(y\left(x\right)\right)}^2+\frac{1}{4x}=0,\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}-2\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\frac{1}{\sqrt{4x{\left(y\left(x\right)\right)}^2+1}}\right)=0,\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}+2\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\frac{1}{\sqrt{4x{\left(y\left(x\right)\right)}^2+1}}\right)=0\}$$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

y(x)^2+1/4/x = 0, ln(x)-_C1+2*arctanh(1/(4*x*y(x)^2+1)^(1/2)) = 0, ln(x)-_C1-2*a
rctanh(1/(4*x*y(x)^2+1)^(1/2)) = 0