ODE
\[ x y(x) y''(x)=2 x y'(x)^2-(y(x)+1) y'(x) \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0655917 (sec), leaf count = 37
\[\left \{\left \{y(x)\to \frac {\tan \left (\frac {\sqrt {c_1} \left (\log (x)-c_2\right )}{\sqrt {2}}\right )}{\sqrt {2} \sqrt {c_1}}\right \}\right \}\]
Maple ✓
cpu = 0.167 (sec), leaf count = 19
\[ \left \{ \ln \left ( x \right ) -2\,{\it \_C1}\,{\it Artanh} \left ( {\frac {y \left ( x \right ) }{{\it \_C1}}} \right ) -{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[x*y[x]*y''[x] == -((1 + y[x])*y'[x]) + 2*x*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> Tan[(Sqrt[C[1]]*(-C[2] + Log[x]))/Sqrt[2]]/(Sqrt[2]*Sqrt[C[1]])}}
Maple raw input
dsolve(x*y(x)*diff(diff(y(x),x),x) = 2*x*diff(y(x),x)^2-(1+y(x))*diff(y(x),x), y(x),'implicit')
Maple raw output
ln(x)-2*_C1*arctanh(1/_C1*y(x))-_C2 = 0