ODE
\[ x y(x) y''(x)=x y'(x)^2-y(x) y'(x) \] ODE Classification
[_Liouville, [_Painleve, `3rd`], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0325922 (sec), leaf count = 12
\[\left \{\left \{y(x)\to c_2 x^{c_1}\right \}\right \}\]
Maple ✓
cpu = 0.015 (sec), leaf count = 15
\[ \left \{ \ln \left ( y \left ( x \right ) \right ) -{\it \_C1}\,\ln \left ( x \right ) -{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[x*y[x]*y''[x] == -(y[x]*y'[x]) + x*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> x^C[1]*C[2]}}
Maple raw input
dsolve(x*y(x)*diff(diff(y(x),x),x) = x*diff(y(x),x)^2-y(x)*diff(y(x),x), y(x),'implicit')
Maple raw output
ln(y(x))-_C1*ln(x)-_C2 = 0