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 = 0.244613 (sec), leaf count = 21
\[\left \{\left \{y(x)\to \frac {c_2}{(a-1) c_1 x+x^a}\right \}\right \}\]
Maple ✓
cpu = 0.064 (sec), leaf count = 31
\[ \left \{ {\frac {a}{xy \left ( x \right ) }}-{\frac {1}{xy \left ( x \right ) }}-{x}^{a-1}{\it \_C1}+{\it \_C2}=0 \right \} \] 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] -> C[2]/(x^a + (-1 + a)*x*C[1])}}
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),'implicit')
Maple raw output
1/x*a/y(x)-1/x/y(x)-x^(a-1)*_C1+_C2 = 0