ODE
\[ x \left (a x y(x)+x^2+2 y(x)^2\right ) y'(x)=y(x)^2 (a x+2 y(x)) \] ODE Classification
[[_homogeneous, `class A`], _rational, _dAlembert]
Book solution method
Homogeneous equation
Mathematica ✓
cpu = 0.0639051 (sec), leaf count = 31
\[\text {Solve}\left [\frac {a y(x)}{x}+\frac {y(x)^2}{x^2}+\log \left (\frac {y(x)}{x}\right )+\log (x)=c_1,y(x)\right ]\]
Maple ✓
cpu = 0.013 (sec), leaf count = 37
\[ \left \{ -{\frac {ay \left ( x \right ) }{x}}-\ln \left ( {\frac {y \left ( x \right ) }{x}} \right ) -{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}-\ln \left ( x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[x*(x^2 + a*x*y[x] + 2*y[x]^2)*y'[x] == y[x]^2*(a*x + 2*y[x]),y[x],x]
Mathematica raw output
Solve[Log[x] + Log[y[x]/x] + (a*y[x])/x + y[x]^2/x^2 == C[1], y[x]]
Maple raw input
dsolve(x*(x^2+a*x*y(x)+2*y(x)^2)*diff(y(x),x) = (a*x+2*y(x))*y(x)^2, y(x),'implicit')
Maple raw output
-a/x*y(x)-ln(y(x)/x)-1/x^2*y(x)^2-ln(x)-_C1 = 0