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 = 0.0301126 (sec), leaf count = 82

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

Maple
cpu = 0.026 (sec), leaf count = 65

\[ \left \{ y \relax (x ) ={\frac {{x}^{2}}{2}}-{\frac {x}{2}\sqrt {{x}^{2}-1}}+{\frac {1}{2}\ln \left (x+\sqrt {{x}^{2}-1} \right ) }+{\it \_C1},y \relax (x ) ={\frac {{x}^{2}}{2}}+{\frac {x}{2}\sqrt {{x}^{2}-1}}-{\frac {1}{2}\ln \left (x+\sqrt {{x}^{2}-1} \right ) }+{\it \_C1} \right \} \] 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),'implicit')

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