4.13.49 \(x \left (x^2-x y(x)+y(x)^2\right ) y'(x)+y(x) \left (x^2+x y(x)+y(x)^2\right )=0\)

ODE
\[ x \left (x^2-x y(x)+y(x)^2\right ) y'(x)+y(x) \left (x^2+x y(x)+y(x)^2\right )=0 \] ODE Classification

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0363929 (sec), leaf count = 26

\[\text {Solve}\left [c_1+\tan ^{-1}\left (\frac {y(x)}{x}\right )=\log \left (\frac {y(x)}{x}\right )+2 \log (x),y(x)\right ]\]

Maple
cpu = 0.018 (sec), leaf count = 29

\[ \left \{ {\frac {1}{2}\arctan \left ({\frac {y \relax (x ) }{x}} \right ) }-{\frac {1}{2}\ln \left ({\frac {y \relax (x ) }{x}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y[x]*(x^2 + x*y[x] + y[x]^2) + x*(x^2 - x*y[x] + y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[ArcTan[y[x]/x] + C[1] == 2*Log[x] + Log[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) = 0, y(x),'implicit')

Maple raw output

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