4.688   x3 + y(x )(3x2 + 2y(x )2)y′(x) = 0

ODE

x3 + y(x)(3x2 + 2y(x)2)y′(x) = 0

ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0653655 (sec), leaf count = 209

{{         ∘ -√------------------------} {        ∘--√-----------------------}  {         ∘√----------------------- } {        ∘√-----------------------} }
           --−--8e2c1x2-+e4c1 +-e2c1-− 4x2          -−---8e2c1x2 +-e4c1-+e2c1 −-4x2           ---8e2c1x2 +-e4c1 +-e2c1 −-4x2         ---8e2c1x2 +-e4c1 +-e2c1 −-4x2
   y(x) → −            2√ 2             ,  y(x) →            2√ 2              , y(x) → −            2√2             ,  y(x ) →            2√2

Maple
cpu = 0.024 (sec), leaf count = 43

{    (  2       2)     (  2         2)                }
 1 ln  x--+-(y2(x))-  − ln  x-+-2(y2(x))-  − ln(x)− -C1 = 0
 2         x                  x

Mathematica raw input

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

Mathematica raw output

{{y[x] -> -Sqrt[E^(2*C[1]) - 4*x^2 - Sqrt[E^(4*C[1]) + 8*E^(2*C[1])*x^2]]/(2*Sqr
t[2])}, {y[x] -> Sqrt[E^(2*C[1]) - 4*x^2 - Sqrt[E^(4*C[1]) + 8*E^(2*C[1])*x^2]]/
(2*Sqrt[2])}, {y[x] -> -Sqrt[E^(2*C[1]) - 4*x^2 + Sqrt[E^(4*C[1]) + 8*E^(2*C[1])
*x^2]]/(2*Sqrt[2])}, {y[x] -> Sqrt[E^(2*C[1]) - 4*x^2 + Sqrt[E^(4*C[1]) + 8*E^(2
*C[1])*x^2]]/(2*Sqrt[2])}}

Maple raw input

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

Maple raw output

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