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]