ODE
\[ x (y(x)+x) y''(x)+x y'(x)^2+(x-y(x)) y'(x)=y(x) \] ODE Classification
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.118935 (sec), leaf count = 79
\[\left \{\left \{y(x)\to -x-\sqrt {-e^{2 c_2} \left (x^2-1\right )+\left (1-2 i c_1\right ) x^2}\right \},\left \{y(x)\to -x+\sqrt {-e^{2 c_2} \left (x^2-1\right )+\left (1-2 i c_1\right ) x^2}\right \}\right \}\]
Maple ✓
cpu = 1.314 (sec), leaf count = 26
\[ \left \{ {\it \_C2}+2\,{\frac {y \left ( x \right ) }{x}}+{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}-{\frac {{\it \_C1}}{{x}^{2}}}=0 \right \} \] Mathematica raw input
DSolve[(x - y[x])*y'[x] + x*y'[x]^2 + x*(x + y[x])*y''[x] == y[x],y[x],x]
Mathematica raw output
{{y[x] -> -x - Sqrt[-(E^(2*C[2])*(-1 + x^2)) + x^2*(1 - (2*I)*C[1])]}, {y[x] -> -x + Sqrt[-(E^(2*C[2])*(-1 + x^2)) + x^2*(1 - (2*I)*C[1])]}}
Maple raw input
dsolve(x*(x+y(x))*diff(diff(y(x),x),x)+x*diff(y(x),x)^2+(x-y(x))*diff(y(x),x) = y(x), y(x),'implicit')
Maple raw output
_C2+2*y(x)/x+1/x^2*y(x)^2-1/x^2*_C1 = 0