Determining stability about a critical point using eigenfunctions

ProdofChem

I admit I am a bit out of practice when it comes to DiffEq. I think I am either forgetting a simple step or getting my methods mixed up.
1. The problem statement, all variables and given/known data
The problem concerns a pendulum defined by

d²θ/dt² + (c/mL)(dθ/dt) + (g/L)sinθ = 0
where m=1, L=1, c=0.5, and of course g=9.8

After converting the DE to a first order system:

set x = θ and y = x' so that
dx/dt = y
dy/dt = -9.8sinx - 0.5y

And identifying critical points:

(n∏,0) where n is an integer

I am asked to linearize the nonlinear system and determine stability about the critical points. If i can get some help with the first point, I should be able to figure out the others.

2. Relevant equations

dx/dt = y
dy/dt = -9.8sinx - 0.5y

3. The attempt at a solution

I have been rummaging through my notes on eigenfunctions and am more or less at a loss how an eigenvalue determines stability. I think I want to take the partial derivatives of the system and evaluate at the point (0,0)...

f_x(dy/dt) = -9.8cosx
f_y(dy/dt) = -0.5
f_x(dx/dt) = 0
f_y(dx/dt) = 1

evaluated @ (0,0):
f_x(dy/dt) = -9.8
f_y(dy/dt) = -0.5
f_x(dx/dt) = 0
f_y(dx/dt) = 1

But I'm not sure why I care or how to proceed to determine the stability about that point.

Clarification: I would like some tips on how to linearize my function and how to get it into the form with which I can determine its eigenvalues ( A-lambda)v=0

vela

Expand sin x as a first-order Taylor polynomial. That's the linear approximation of sin x.