4.18.15 $x{y}^{\prime }(x{)}^2+(x-y(x))y^{\prime }(x)-y(x)+1=0$

ODE
$$x{y}^{\prime }(x{)}^2+(x-y(x))y^{\prime }(x)-y(x)+1=0$$ 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 = 0.311527 (sec), leaf count = 16

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

Maple ✓
cpu = 0.03 (sec), leaf count = 38

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

DSolve[1 - y[x] + (x - y[x])*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> x*(-1 + C[1]^(-1)) + C[1]}}

Maple raw input

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

Maple raw output

y(x)^2+2*x*y(x)+x^2-4*x = 0, y(x) = (_C1^2*x+_C1*x+1)/(_C1+1)