##### 4.13.50 $$x \left (x^2-x y(x)-y(x)^2\right ) y'(x)=y(x) \left (x^2+x y(x)-y(x)^2\right )$$

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

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

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.522389 (sec), leaf count = 30

$\text {Solve}\left [c_1=\frac {x}{y(x)}+\frac {y(x)}{x}+\log \left (\frac {y(x)}{x}\right )+2 \log (x),y(x)\right ]$

Maple
cpu = 0.354 (sec), leaf count = 29

$[y \left (x \right ) = {\mathrm e}^{\RootOf \left (2 \,{\mathrm e}^{\textit {\_Z}} \ln \left (x \right )+{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} \textit {\_C1} +{\mathrm e}^{\textit {\_Z}} \textit {\_Z} +1\right )} x]$ Mathematica raw input

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

Mathematica raw output

Solve[C[1] == 2*Log[x] + Log[y[x]/x] + x/y[x] + y[x]/x, y[x]]

Maple raw input

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

Maple raw output

[y(x) = exp(RootOf(2*exp(_Z)*ln(x)+exp(_Z)^2+2*exp(_Z)*_C1+exp(_Z)*_Z+1))*x]