##### 4.16.43 $$y'(x)^2-2 x y'(x)+1=0$$

ODE
$y'(x)^2-2 x y'(x)+1=0$ ODE Classiﬁcation

[_quadrature]

Book solution method
Missing Variables ODE, Dependent variable missing, Solve for $$x$$

Mathematica
cpu = 0.166622 (sec), leaf count = 82

$\left \{\left \{y(x)\to \frac {1}{2} \left (x^2+\sqrt {x^2-1} x-\log \left (\sqrt {x^2-1}+x\right )+2 c_1\right )\right \},\left \{y(x)\to \frac {1}{2} \left (x^2-\sqrt {x^2-1} x+\log \left (\sqrt {x^2-1}+x\right )+2 c_1\right )\right \}\right \}$

Maple
cpu = 0.036 (sec), leaf count = 65

$\left [y \left (x \right ) = \frac {x^{2}}{2}-\frac {\sqrt {x^{2}-1}\, x}{2}+\frac {\ln \left (x +\sqrt {x^{2}-1}\right )}{2}+\textit {\_C1}, y \left (x \right ) = \frac {x^{2}}{2}+\frac {\sqrt {x^{2}-1}\, x}{2}-\frac {\ln \left (x +\sqrt {x^{2}-1}\right )}{2}+\textit {\_C1}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (x^2 + x*Sqrt[-1 + x^2] + 2*C[1] - Log[x + Sqrt[-1 + x^2]])/2}, {y[x]
-> (x^2 - x*Sqrt[-1 + x^2] + 2*C[1] + Log[x + Sqrt[-1 + x^2]])/2}}

Maple raw input

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

Maple raw output

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