##### 4.16.28 $$y'(x)^2-2 y'(x)-y(x)^2=0$$

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

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Solve for $$y'$$

Mathematica
cpu = 0.228742 (sec), leaf count = 73

$\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\text {\#1}^2+1}}{\text {\#1}}-\frac {1}{\text {\#1}}+\sinh ^{-1}(\text {\#1})\& \right ][-x+c_1]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\text {\#1}^2+1}}{\text {\#1}}+\frac {1}{\text {\#1}}+\sinh ^{-1}(\text {\#1})\& \right ][x+c_1]\right \}\right \}$

Maple
cpu = 0.037 (sec), leaf count = 85

$\left [x -\frac {1}{y \left (x \right )}-\frac {\left (1+y \left (x \right )^{2}\right )^{\frac {3}{2}}}{y \left (x \right )}+y \left (x \right ) \sqrt {1+y \left (x \right )^{2}}+\arcsinh \left (y \left (x \right )\right )-\textit {\_C1} = 0, x +\frac {\left (1+y \left (x \right )^{2}\right )^{\frac {3}{2}}}{y \left (x \right )}-y \left (x \right ) \sqrt {1+y \left (x \right )^{2}}-\arcsinh \left (y \left (x \right )\right )-\frac {1}{y \left (x \right )}-\textit {\_C1} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> InverseFunction[ArcSinh[#1] - #1^(-1) - Sqrt[1 + #1^2]/#1 & ][-x + C[1
]]}, {y[x] -> InverseFunction[ArcSinh[#1] + #1^(-1) - Sqrt[1 + #1^2]/#1 & ][x +
C[1]]}}

Maple raw input

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

Maple raw output

[x-1/y(x)-1/y(x)*(1+y(x)^2)^(3/2)+y(x)*(1+y(x)^2)^(1/2)+arcsinh(y(x))-_C1 = 0, x
+1/y(x)*(1+y(x)^2)^(3/2)-y(x)*(1+y(x)^2)^(1/2)-arcsinh(y(x))-1/y(x)-_C1 = 0]