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 = 0.133746 (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 = 0.018 (sec), leaf count = 34

\[ \left \{ -{\frac {1}{2}\ln \left ({\frac {y \relax (x ) }{x}} \right ) }-{\frac {x}{2\,y \relax (x ) }}-{\frac {y \relax (x ) }{2\,x}}-\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

-1/2*ln(y(x)/x)-1/2*x/y(x)-1/2*y(x)/x-ln(x)-_C1 = 0