##### 4.14.38 $$x^3+y(x) \left (3 x^2+2 y(x)^2\right ) y'(x)=0$$

ODE
$x^3+y(x) \left (3 x^2+2 y(x)^2\right ) y'(x)=0$ ODE Classiﬁcation

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

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.449944 (sec), leaf count = 209

$\left \{\left \{y(x)\to -\frac {\sqrt {-4 x^2-\sqrt {8 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}}}{2 \sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {-4 x^2-\sqrt {8 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}}}{2 \sqrt {2}}\right \},\left \{y(x)\to -\frac {\sqrt {-4 x^2+\sqrt {8 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}}}{2 \sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {-4 x^2+\sqrt {8 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}}}{2 \sqrt {2}}\right \}\right \}$

Maple
cpu = 0.301 (sec), leaf count = 137

$\left [y \left (x \right ) = -\frac {\sqrt {-8 \textit {\_C1}^{2} x^{2}-2 \sqrt {8 \textit {\_C1}^{2} x^{2}+1}+2}}{4 \textit {\_C1}}, y \left (x \right ) = \frac {\sqrt {-8 \textit {\_C1}^{2} x^{2}-2 \sqrt {8 \textit {\_C1}^{2} x^{2}+1}+2}}{4 \textit {\_C1}}, y \left (x \right ) = -\frac {\sqrt {-8 \textit {\_C1}^{2} x^{2}+2 \sqrt {8 \textit {\_C1}^{2} x^{2}+1}+2}}{4 \textit {\_C1}}, y \left (x \right ) = \frac {\sqrt {-8 \textit {\_C1}^{2} x^{2}+2 \sqrt {8 \textit {\_C1}^{2} x^{2}+1}+2}}{4 \textit {\_C1}}\right ]$ 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] -> -1/2*Sqrt[E^(2*C[1]) - 4*x^2 - Sqrt[E^(4*C[1]) + 8*E^(2*C[1])*x^2]]/Sq
rt[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] -> -1/2*Sqrt[E^(2*C[1]) - 4*x^2 + Sqrt[E^(4*C[1]) + 8*E^(2*C
[1])*x^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))

Maple raw output

[y(x) = -1/4/_C1*(-8*_C1^2*x^2-2*(8*_C1^2*x^2+1)^(1/2)+2)^(1/2), y(x) = 1/4/_C1*
(-8*_C1^2*x^2-2*(8*_C1^2*x^2+1)^(1/2)+2)^(1/2), y(x) = -1/4/_C1*(-8*_C1^2*x^2+2*
(8*_C1^2*x^2+1)^(1/2)+2)^(1/2), y(x) = 1/4/_C1*(-8*_C1^2*x^2+2*(8*_C1^2*x^2+1)^(
1/2)+2)^(1/2)]