ODE
\[ a y(x) y'(x)+x y(x) y''(x)-2 x y'(x)^2=0 \] ODE Classification
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.439708 (sec), leaf count = 24
\[\left \{\left \{y(x)\to \frac {c_2 x^a}{x+(a-1) c_1 x^a}\right \}\right \}\]
Maple ✓
cpu = 0.254 (sec), leaf count = 31
\[\left [y \left (x \right ) = -\frac {x^{a} \left (a -1\right )}{\textit {\_C2} \,x^{a} a -\textit {\_C2} \,x^{a}-\textit {\_C1} x}\right ]\] Mathematica raw input
DSolve[a*y[x]*y'[x] - 2*x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x^a*C[2])/(x + (-1 + a)*x^a*C[1])}}
Maple raw input
dsolve(x*y(x)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)^2+a*y(x)*diff(y(x),x) = 0, y(x))
Maple raw output
[y(x) = -x^a*(a-1)/(_C2*x^a*a-_C2*x^a-_C1*x)]