##### 4.11.37 $$x \left (x^3+y(x)\right ) y'(x)=\left (x^3-y(x)\right ) y(x)$$

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

[[_homogeneous, class G], _rational, [_Abel, 2nd type, class B]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.276046 (sec), leaf count = 68

$\left \{\left \{y(x)\to \frac {x^4}{-x+\frac {\sqrt {1+c_1 x^4}}{\sqrt {\frac {1}{x^2}}}}\right \},\left \{y(x)\to -\frac {x^4}{x+\frac {\sqrt {1+c_1 x^4}}{\sqrt {\frac {1}{x^2}}}}\right \}\right \}$

Maple
cpu = 0.242 (sec), leaf count = 41

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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