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))]