ODE
\[ x^2 y(x) y''(x)+\left (x y'(x)-y(x)\right )^2=0 \] 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 = 0.0379801 (sec), leaf count = 23
\[\left \{\left \{y(x)\to c_2 \sqrt {x} \sqrt {c_1+2 x}\right \}\right \}\]
Maple ✓
cpu = 0.069 (sec), leaf count = 18
\[ \left \{ -{\it \_C1}\,x+{\it \_C2}-{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{2\,x}}=0 \right \} \] Mathematica raw input
DSolve[(-y[x] + x*y'[x])^2 + x^2*y[x]*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> Sqrt[x]*Sqrt[2*x + C[1]]*C[2]}}
Maple raw input
dsolve(x^2*y(x)*diff(diff(y(x),x),x)+(x*diff(y(x),x)-y(x))^2 = 0, y(x),'implicit')
Maple raw output
-_C1*x+_C2-1/2*y(x)^2/x = 0