A simple gravity pendulum is the simplified model for a pendulum: a point mass suspended from a massless cord suspended from a pivot in a vacuum subject to gravity.
In my class, I studied very simplified spring mass systems where the solution to the second order linear homogeneous equation $y''+C_1y'+C_2y=0$ was the position function of the point mass attached to the spring (where $C_1$ and $C_2$ are the damping constant and the spring constant respectively).
I was told that a simple gravity pendulum could be modeled with a similar second order differential equation, and I would appreciate if someone could derive this equation in a didactic and systematic manner, so I could fully appreciate and understand this model (which, I know, is not very realistic, but interesting to me nonetheless). Thank you.