How to add drag to a simple pendulum problem function ad hoc?

Question

I recently solved (non-numerically) the differential equation of the simple pendulum problem,

$$y''=-c*\sin(y)\tag{1}.$$

I then tried to solve the equation with drag. I failed. I don't want to solve it numerically because I'm doing this for fun and that would be fun, $$y''=-c*sin(y)-a*y'\tag{2}.$$

So how would I go about adding drag to $y$ ad hoc, or without actually solving the second differential equation?

The constants of $y$ in my solution shift the equation along the $x$-axis and change the slope at $y(x)=0$ & $|y'(x)|=y'(x)$.

If you give me an equation of how the amplitude should change over time that would also work since the slope and amplitude are directly related.

Answer 1

I recently solved (non-numerically) the differential equation of the simple pendulum problem.

As @G. wrote in the comments:

Please share your non-numerical solution to the first equation.

The ODE:

$$y''=-c \sin(y)$$

is usually solved with the small angle approximation, which is assumed valid for small angles $y$:

$$\sin(y) \approx y$$

So the ODE becomes:

$$y''=-cy$$

which does have a simple analytical solution:

$$y=C_1\sin(\sqrt{c}x)+C_2\cos(\sqrt{c}x)$$

where $C_1$ and $C_2$ are integration constants, determined from boundary/initial conditions.

Now if we add the damping term $ay'$, slight reworking gives:

$$y''+ay'+cy=0$$

This is a second order linear homogeneous ODE which also has analytical solutions. You can find the algorithm on plenty web resources, just google it.