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 = 0.866326 (sec), leaf count = 37
\[\left \{\left \{y(x)\to x \left (\left (\frac {(-1)^{-a} (a+1) \left ((-1)^a c_2 x+c_1\right )}{x}\right ){}^{\frac {1}{a+1}}+1\right )\right \}\right \}\]
Maple ✓
cpu = 0.078 (sec), leaf count = 38
\[ \left \{ \left ( y \left ( x \right ) {x}^{-a}-{x}^{1-a} \right ) \left ( x-y \left ( x \right ) \right ) ^{a}+x \left ( 1+a \right ) {\it \_C2}+{\it \_C1}=0 \right \} \] 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),'implicit')
Maple raw output
(y(x)*x^(-a)-x^(1-a))*(x-y(x))^a+x*(1+a)*_C2+_C1 = 0