4.11.37 $x(x^3+y(x))y^{\prime }(x)=(x^3-y(x))y(x)$

ODE
$$x(x^3+y(x))y^{\prime }(x)=(x^3-y(x))y(x)$$ ODE Classification

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

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.0108369 (sec), leaf count = 68

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

Maple ✓
cpu = 0.02 (sec), leaf count = 33

$$\{\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}+\frac{1}{2}\ln \left(\frac{y\left(x\right)}{x^3}\right)-\frac{1}{4}\ln \left(\frac{x^3+2y\left(x\right)}{x^3}\right)=0\}$$ 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),'implicit')

Maple raw output

ln(x)-_C1+1/2*ln(y(x)/x^3)-1/4*ln((x^3+2*y(x))/x^3) = 0