ODE
\[ x^2 (x-y(x)) y''(x)=\left (x y'(x)-y(x)\right )^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0696101 (sec), leaf count = 56
\[\left \{\left \{y(x)\to x-\sqrt {x} \sqrt {2 c_2 x-2 c_1+x}\right \},\left \{y(x)\to \sqrt {x} \sqrt {2 c_2 x-2 c_1+x}+x\right \}\right \}\]
Maple ✓
cpu = 0.039 (sec), leaf count = 21
\[ \left \{ x{\it \_C1}+{\it \_C2}-y \left ( x \right ) +{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{2\,x}}=0 \right \} \] Mathematica raw input
DSolve[x^2*(x - y[x])*y''[x] == (-y[x] + x*y'[x])^2,y[x],x]
Mathematica raw output
{{y[x] -> x - Sqrt[x]*Sqrt[x - 2*C[1] + 2*x*C[2]]}, {y[x] -> x + Sqrt[x]*Sqrt[x - 2*C[1] + 2*x*C[2]]}}
Maple raw input
dsolve(x^2*(x-y(x))*diff(diff(y(x),x),x) = (x*diff(y(x),x)-y(x))^2, y(x),'implicit')
Maple raw output
x*_C1+_C2-y(x)+1/2*y(x)^2/x = 0