When the angle of swing is not small, the motion of a pendulum is no longer well approximated by simple harmonic motion. The solution, by way of the elliptic integral, cannot be expressed in a finite number of terms of elementary functions.
However, a differential equation is a complete model of a problem. I think, this is what Newton was saying.
Writing the problem as finite differences, we can examine it numerically.... or pass it to a computer. Integration in time = Press play.

I assume the mass of the rod is negligible. Say, a rigid wire.
The difference equation is in the update script of t. It is as follows:
Given step n-1:
θ[n-1] = (the current angle out-of-rest of the pendulum, in radians)
v[n-1] = (current velocity)
F[n-1] = (the component of gravitational force affecting the pendulum at its current position).
Step n:
Δv[n] = (F[n-1]/m) Δt
v[n] = v[n-1]+Δv[n]
Δθ[n] = (v[n]\l)Δt
θ[n] =θ[n-1] + Δθ[n]
Update: Added damping term μ, proportional to velocity. The update script is the same, but F[n] includes a second term,
F[n] = F1 + F2 = −mg sinθ −2μmg v