4.17.31 \(2 x y(x)^3 y'(x)+y'(x)^2+y(x)^4=0\)

ODE
\[ 2 x y(x)^3 y'(x)+y'(x)^2+y(x)^4=0 \] ODE Classification

[[_homogeneous, `class G`]]

Book solution method
No Missing Variables ODE, Solve for \(x\)

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

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

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

\[ \left \{ \left (y \relax (x ) \right ) ^{2}-{x}^{-2}=0,\ln \relax (x ) -{\it \_C1}-\ln \left (xy \relax (x ) +\sqrt {{x}^{2} \left (y \relax (x ) \right ) ^{2}-1} \right ) -\ln \left (xy \relax (x ) \right ) =0,\ln \relax (x ) -{\it \_C1}+\ln \left (xy \relax (x ) +\sqrt {{x}^{2} \left (y \relax (x ) \right ) ^{2}-1} \right ) -\ln \left (xy \relax (x ) \right ) =0 \right \} \] 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