ODE
\[ (x-y(x)) y''(x)+2 y'(x) \left (y'(x)+1\right )=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.225421 (sec), leaf count = 27
\[\left \{\left \{y(x)\to -\frac {c_2 \left (c_2+x\right )+e^{-c_1}}{c_2+x}\right \}\right \}\]
Maple ✓
cpu = 1.368 (sec), leaf count = 63
\[ \left \{ -{\it \_C2}+x+y \left ( x \right ) -\sqrt { \left ( x-y \left ( x \right ) -2\,{\it \_C1} \right ) \left ( x-y \left ( x \right ) +2\,{\it \_C1} \right ) }=0,-{\it \_C2}+x+y \left ( x \right ) +\sqrt { \left ( x-y \left ( x \right ) -2\,{\it \_C1} \right ) \left ( x-y \left ( x \right ) +2\,{\it \_C1} \right ) }=0 \right \} \] Mathematica raw input
DSolve[2*y'[x]*(1 + y'[x]) + (x - y[x])*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((E^(-C[1]) + C[2]*(x + C[2]))/(x + C[2]))}}
Maple raw input
dsolve((x-y(x))*diff(diff(y(x),x),x)+2*diff(y(x),x)*(1+diff(y(x),x)) = 0, y(x),'implicit')
Maple raw output
-_C2+x+y(x)-((x-y(x)-2*_C1)*(x-y(x)+2*_C1))^(1/2) = 0, -_C2+x+y(x)+((x-y(x)-2*_C1)*(x-y(x)+2*_C1))^(1/2) = 0