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 = 0.0108369 (sec), leaf count = 68
\[\left \{\left \{y(x)\to \frac {x^4}{\frac {\sqrt {c_1 x^4+1}}{\sqrt {\frac {1}{x^2}}}-x}\right \},\left \{y(x)\to -\frac {x^4}{\frac {\sqrt {c_1 x^4+1}}{\sqrt {\frac {1}{x^2}}}+x}\right \}\right \}\]
Maple ✓
cpu = 0.02 (sec), leaf count = 33
\[ \left \{ \ln \left ( x \right ) -{\it \_C1}+{\frac {1}{2}\ln \left ( {\frac {y \left ( x \right ) }{{x}^{3}}} \right ) }-{\frac {1}{4}\ln \left ( {\frac {{x}^{3}+2\,y \left ( x \right ) }{{x}^{3}}} \right ) }=0 \right \} \] 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),'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