ODE
\[ x^2 (x-y(x)) y''(x)=a \left (x y'(x)-y(x)\right )^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.808152 (sec), leaf count = 37
\[\left \{\left \{y(x)\to x \left (1+\left (\frac {(-1)^{-a} (a+1) \left ((-1)^a c_2 x+c_1\right )}{x}\right ){}^{\frac {1}{a+1}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.921 (sec), leaf count = 36
\[[y \left (x \right ) = -\RootOf \left (-\textit {\_Z}^{1+a} x^{-a}+\textit {\_C1} a x +\textit {\_C1} x -\textit {\_C2} a -\textit {\_C2} \right )+x]\] Mathematica raw input
DSolve[x^2*(x - y[x])*y''[x] == a*(-y[x] + x*y'[x])^2,y[x],x]
Mathematica raw output
{{y[x] -> x*(1 + (((1 + a)*(C[1] + (-1)^a*x*C[2]))/((-1)^a*x))^(1 + a)^(-1))}}
Maple raw input
dsolve(x^2*(x-y(x))*diff(diff(y(x),x),x) = a*(x*diff(y(x),x)-y(x))^2, y(x))
Maple raw output
[y(x) = -RootOf(-_Z^(1+a)*x^(-a)+_C1*a*x+_C1*x-_C2*a-_C2)+x]