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 = 0.151031 (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 = 0.02 (sec), leaf count = 26
\[ \left \{ -{\it \_C1}\,x{x}^{a}-{\it \_C2}+{\frac {y \left ( x \right ) \left ( y \left ( x \right ) \right ) ^{a}}{1+a}}=0 \right \} \] 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),'implicit')
Maple raw output
-_C1*x*x^a-_C2+y(x)/(1+a)*y(x)^a = 0