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...