ODE
\[ x^2 y(x) y''(x)=a x y(x) y'(x)+a y(x)^2+2 x^2 y'(x)^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.27606 (sec), leaf count = 26
\[\left \{\left \{y(x)\to \frac {a-1}{(a-1) c_2 x^a-c_1 x}\right \}\right \}\]
Maple ✓
cpu = 0.404 (sec), leaf count = 24
\[\left [y \left (x \right ) = \frac {a -1}{x \left (x^{a -1} \textit {\_C1} -\textit {\_C2} \right )}\right ]\] Mathematica raw input
DSolve[x^2*y[x]*y''[x] == a*y[x]^2 + a*x*y[x]*y'[x] + 2*x^2*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> (-1 + a)/(-(x*C[1]) + (-1 + a)*x^a*C[2])}}
Maple raw input
dsolve(x^2*y(x)*diff(diff(y(x),x),x) = 2*x^2*diff(y(x),x)^2+a*x*y(x)*diff(y(x),x)+a*y(x)^2, y(x))
Maple raw output
[y(x) = (a-1)/x/(x^(a-1)*_C1-_C2)]