##### 4.14.34 $$y(x) \left (3 x^2+y(x)^2\right ) y'(x)+x \left (x^2+3 y(x)^2\right )=0$$

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

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

Book solution method
Exact equation

Mathematica
cpu = 0.45869 (sec), leaf count = 129

$\left \{\left \{y(x)\to -\sqrt {-3 x^2-\sqrt {8 x^4+e^{4 c_1}}}\right \},\left \{y(x)\to \sqrt {-3 x^2-\sqrt {8 x^4+e^{4 c_1}}}\right \},\left \{y(x)\to -\sqrt {-3 x^2+\sqrt {8 x^4+e^{4 c_1}}}\right \},\left \{y(x)\to \sqrt {-3 x^2+\sqrt {8 x^4+e^{4 c_1}}}\right \}\right \}$

Maple
cpu = 0.249 (sec), leaf count = 119

$\left [y \left (x \right ) = \frac {\sqrt {-3 x^{2} \textit {\_C1} -\sqrt {8 x^{4} \textit {\_C1}^{2}+1}}}{\sqrt {\textit {\_C1}}}, y \left (x \right ) = \frac {\sqrt {-3 x^{2} \textit {\_C1} +\sqrt {8 x^{4} \textit {\_C1}^{2}+1}}}{\sqrt {\textit {\_C1}}}, y \left (x \right ) = -\frac {\sqrt {-3 x^{2} \textit {\_C1} -\sqrt {8 x^{4} \textit {\_C1}^{2}+1}}}{\sqrt {\textit {\_C1}}}, y \left (x \right ) = -\frac {\sqrt {-3 x^{2} \textit {\_C1} +\sqrt {8 x^{4} \textit {\_C1}^{2}+1}}}{\sqrt {\textit {\_C1}}}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -Sqrt[-3*x^2 - Sqrt[E^(4*C[1]) + 8*x^4]]}, {y[x] -> Sqrt[-3*x^2 - Sqrt
[E^(4*C[1]) + 8*x^4]]}, {y[x] -> -Sqrt[-3*x^2 + Sqrt[E^(4*C[1]) + 8*x^4]]}, {y[x
] -> Sqrt[-3*x^2 + Sqrt[E^(4*C[1]) + 8*x^4]]}}

Maple raw input

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

Maple raw output

[y(x) = (-3*x^2*_C1-(8*_C1^2*x^4+1)^(1/2))^(1/2)/_C1^(1/2), y(x) = (-3*x^2*_C1+(
8*_C1^2*x^4+1)^(1/2))^(1/2)/_C1^(1/2), y(x) = -(-3*x^2*_C1-(8*_C1^2*x^4+1)^(1/2)
)^(1/2)/_C1^(1/2), y(x) = -(-3*x^2*_C1+(8*_C1^2*x^4+1)^(1/2))^(1/2)/_C1^(1/2)]