4.18.31 $(x+1)y^{\prime }(x{)}^2-(y(x)+x)y^{\prime }(x)+y(x)=0$

ODE
$$(x+1)y^{\prime }(x{)}^2-(y(x)+x)y^{\prime }(x)+y(x)=0$$ ODE Classification

[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]

Book solution method
Clairaut’s equation and related types, $f(y-x{y}^{\prime },y^{\prime })=0$

Mathematica ✓
cpu = 0.256356 (sec), leaf count = 57

$$\{\{y(x)\to -\frac{e^{c_1}(e^{c_1}-2x)}{2(e^{c_1}+2)}\},\{y(x)\to \frac{2e^{c_1}(x-2e^{c_1})}{2e^{c_1}+1}\}\}$$

Maple ✓
cpu = 0.033 (sec), leaf count = 37

$$\{{\left(y\left(x\right)\right)}^2+(-2x-4)y\left(x\right)+x^2=0,y\left(x\right)=\frac{\mathit{\_}\mathit{C}\mathit{1}(\mathit{\_}\mathit{C}\mathit{1}x+\mathit{\_}\mathit{C}\mathit{1}-x)}{\mathit{\_}\mathit{C}\mathit{1}-1}\}$$ 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)