4.19.38 $x^4{y}^{\prime }(x{)}^2+2x^{3}y(x)y^{\prime }(x)-4=0$

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

[[_homogeneous, `class G`], _rational]

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

Mathematica ✓
cpu = 0.630521 (sec), leaf count = 49

$$\{\{y(x)\to \frac{e^{c_1}}{2x^2}-2e^{-c_1}\},\{y(x)\to \frac{e^{-c_1}}{2}-\frac{2e^{c_1}}{x^2}\}\}$$

Maple ✓
cpu = 0.07 (sec), leaf count = 43

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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