Converting nonlinear second order ODE to first order The second order ODE is: 
$$
y'' + ky' + (mg/c)\sin(y) = 0
$$
$k$, $m$, $g$ and $c$ and constants.

What is the first order ODE and how do I do it?
Let $x_1 = y$ and $x_2 = \dot{y}$. Then $\dot{x}_1 = x_2$ and $\dot{x}_2 = -kx_2-\frac{mg}{c}\sin(x_1)$. Now stack $x_1$ and $x_2$ into a column vector, i.e. $x = [x_1 \ x_2]^\top$. Then, $\dot{x} = f(x)$ where $f(x)$ is a vector valued function which follows from what I wrote above. This is the first order form.
If y is "small" then sin y ~y
This called linearzation

This is special case .