ODE
\[ (x+1) y'(x)^2-(y(x)+x) y'(x)+y(x)=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 0.256356 (sec), leaf count = 57
\[\left \{\left \{y(x)\to -\frac {e^{c_1} \left (e^{c_1}-2 x\right )}{2 \left (e^{c_1}+2\right )}\right \},\left \{y(x)\to \frac {2 e^{c_1} \left (x-2 e^{c_1}\right )}{2 e^{c_1}+1}\right \}\right \}\]
Maple ✓
cpu = 0.033 (sec), leaf count = 37
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}+ \left ( -2\,x-4 \right ) y \left ( x \right ) +{x}^{2}=0,y \left ( x \right ) ={\frac {{\it \_C1}\, \left ( {\it \_C1}\,x+{\it \_C1}-x \right ) }{{\it \_C1}-1}} \right \} \] Mathematica raw input
DSolve[y[x] - (x + y[x])*y'[x] + (1 + x)*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(E^C[1]*(E^C[1] - 2*x))/(2*(2 + E^C[1]))}, {y[x] -> (2*E^C[1]*(-2*E^C[1] + x))/(1 + 2*E^C[1])}}
Maple raw input
dsolve((1+x)*diff(y(x),x)^2-(x+y(x))*diff(y(x),x)+y(x) = 0, y(x),'implicit')
Maple raw output
y(x)^2+(-2*x-4)*y(x)+x^2 = 0, y(x) = _C1*(_C1*x+_C1-x)/(_C1-1)