4.17.31 $2xy(x{)}^3{y}^{\prime }(x)+y^{\prime }(x{)}^2+y(x{)}^4=0$

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

[[_homogeneous, `class G`]]

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

Mathematica ✓
cpu = 0.773611 (sec), leaf count = 199

$$\{\text{Solve}[\frac{\sqrt{x^{2}y(x{)}^2-1}y(x{)}^2(c_1+\log (y(x)))+\sqrt{y(x{)}^4(x^{2}y(x{)}^2-1)}(\log (y(x))-\log (y(x)(\sqrt{x^{2}y(x{)}^2-1}+xy(x))))}{y(x)\sqrt{x^{2}y(x{)}^2-1}}=0,y(x)],\text{Solve}[\frac{\sqrt{x^{2}y(x{)}^2-1}y(x{)}^2(c_1+\log (y(x)))+\sqrt{y(x{)}^4(x^{2}y(x{)}^2-1)}(\log (y(x)(\sqrt{x^{2}y(x{)}^2-1}+xy(x)))-\log (y(x)))}{y(x)\sqrt{x^{2}y(x{)}^2-1}}=0,y(x)]\}$$

Maple ✓
cpu = 0.075 (sec), leaf count = 81

$$\{{\left(y\left(x\right)\right)}^2-x^{-2}=0,\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}-\ln (xy\left(x\right)+\sqrt{x^2{\left(y\left(x\right)\right)}^2-1})-\ln \left(xy\left(x\right)\right)=0,\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}+\ln (xy\left(x\right)+\sqrt{x^2{\left(y\left(x\right)\right)}^2-1})-\ln \left(xy\left(x\right)\right)=0\}$$ Mathematica raw input

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

Mathematica raw output

{Solve[((C[1] + Log[y[x]])*y[x]^2*Sqrt[-1 + x^2*y[x]^2] + (Log[y[x]] - Log[y[x]*
(x*y[x] + Sqrt[-1 + x^2*y[x]^2])])*Sqrt[y[x]^4*(-1 + x^2*y[x]^2)])/(y[x]*Sqrt[-1
 + x^2*y[x]^2]) == 0, y[x]], Solve[((C[1] + Log[y[x]])*y[x]^2*Sqrt[-1 + x^2*y[x]
^2] + (-Log[y[x]] + Log[y[x]*(x*y[x] + Sqrt[-1 + x^2*y[x]^2])])*Sqrt[y[x]^4*(-1 
+ x^2*y[x]^2)])/(y[x]*Sqrt[-1 + x^2*y[x]^2]) == 0, y[x]]}

Maple raw input

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

Maple raw output

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