ODE
\[ x^3 y''(x)=a \left (x y'(x)-y(x)\right )^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.23202 (sec), leaf count = 25
\[\left \{\left \{y(x)\to -\frac {x \log \left (-\frac {a (c_2 x+c_1)}{x}\right )}{a}\right \}\right \}\]
Maple ✓
cpu = 0.645 (sec), leaf count = 23
\[\left [y \left (x \right ) = -\frac {\ln \left (\frac {a \left (\textit {\_C1} x -\textit {\_C2} \right )}{x}\right ) x}{a}\right ]\] Mathematica raw input
DSolve[x^3*y''[x] == a*(-y[x] + x*y'[x])^2,y[x],x]
Mathematica raw output
{{y[x] -> -((x*Log[-((a*(C[1] + x*C[2]))/x)])/a)}}
Maple raw input
dsolve(x^3*diff(diff(y(x),x),x) = a*(x*diff(y(x),x)-y(x))^2, y(x))
Maple raw output
[y(x) = -ln(a*(_C1*x-_C2)/x)*x/a]