I've already derived the equation, and simplified it to the linear case, now I have to solve it... The equation is:
Where l is length of the pendulum.
Doing some research, I keep finding that the solution is:
Where is the initial condition at time t=0
Now, here's my attempt at a solution:
Since it is linear and homogeneous, we can assume a solution:
Where k is an arbitrary constant.
This leads to the indicial equation:
The solution to which is:
This leads to two solutions:
A linear combination of these leads to the general homogeneous solution:
Expanding the exponentials into their trigonometric form, we get:
From here I'm not sure how to proceed...