##### 4.18.31 $$(x+1) y'(x)^2-(y(x)+x) y'(x)+y(x)=0$$

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 = 0.155201 (sec), leaf count = 19

$\left \{\left \{y(x)\to c_1 \left (x+\frac {c_1}{-1+c_1}\right )\right \}\right \}$

Maple
cpu = 0.041 (sec), leaf count = 59

$\left [y \left (x \right ) = x +2-2 \sqrt {x +1}, y \left (x \right ) = x +2+2 \sqrt {x +1}, y \left (x \right ) = \frac {\left (-\textit {\_C1}^{2}+\textit {\_C1} \right ) x}{1-\textit {\_C1}}-\frac {\textit {\_C1}^{2}}{1-\textit {\_C1}}\right ]$ 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] -> C[1]*(x + C[1]/(-1 + C[1]))}}

Maple raw input

dsolve((x+1)*diff(y(x),x)^2-(x+y(x))*diff(y(x),x)+y(x) = 0, y(x))

Maple raw output

[y(x) = x+2-2*(x+1)^(1/2), y(x) = x+2+2*(x+1)^(1/2), y(x) = (-_C1^2+_C1)/(1-_C1)
*x-_C1^2/(1-_C1)]