ODE
\[ x (x-a y(x)) y'(x)=y(x) (y(x)-a x) \] ODE Classification
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]
Book solution method
Homogeneous equation
Mathematica ✓
cpu = 0.0312607 (sec), leaf count = 34
\[\text {Solve}\left [(a-1) \log \left (1-\frac {y(x)}{x}\right )+(a+1) \log (x)+\log \left (\frac {y(x)}{x}\right )=c_1,y(x)\right ]\]
Maple ✓
cpu = 0.023 (sec), leaf count = 51
\[ \left \{ {\frac {1}{1+a} \left ( \left ( 1-a \right ) \ln \left ( {\frac {y \left ( x \right ) -x}{x}} \right ) -\ln \left ( {\frac {y \left ( x \right ) }{x}} \right ) + \left ( -a-1 \right ) \ln \left ( x \right ) -{\it \_C1}\,a-{\it \_C1} \right ) }=0 \right \} \] Mathematica raw input
DSolve[x*(x - a*y[x])*y'[x] == y[x]*(-(a*x) + y[x]),y[x],x]
Mathematica raw output
Solve[(1 + a)*Log[x] + Log[y[x]/x] + (-1 + a)*Log[1 - y[x]/x] == C[1], y[x]]
Maple raw input
dsolve(x*(x-a*y(x))*diff(y(x),x) = y(x)*(y(x)-a*x), y(x),'implicit')
Maple raw output
((1-a)*ln((y(x)-x)/x)-ln(y(x)/x)+(-a-1)*ln(x)-_C1*a-_C1)/(1+a) = 0