ODE
\[ a y'(x) \left (x y'(x)-y(x)\right )+x y(x) y''(x)=0 \] 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 = 0.444299 (sec), leaf count = 22
\[\left \{\left \{y(x)\to c_2 \left (x^{a+1}+c_1\right ){}^{\frac {1}{a+1}}\right \}\right \}\]
Maple ✓
cpu = 0.601 (sec), leaf count = 26
\[\left [y \left (x \right ) = \left (\frac {1}{\textit {\_C1} \,x^{1+a}+\textit {\_C2} a +\textit {\_C2}}\right )^{-\frac {1}{1+a}}\right ]\] Mathematica raw input
DSolve[a*y'[x]*(-y[x] + x*y'[x]) + x*y[x]*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x^(1 + a) + C[1])^(1 + a)^(-1)*C[2]}}
Maple raw input
dsolve(x*y(x)*diff(diff(y(x),x),x)+a*diff(y(x),x)*(x*diff(y(x),x)-y(x)) = 0, y(x))
Maple raw output
[y(x) = 1/((1/(_C1*x^(1+a)+_C2*a+_C2))^(1/(1+a)))]