We begin our study of the numerical solution of ordinary differential equations by modeling a mass on a spring using a Hooke's Law approximation to the spring force. Although the analytic solution to this simple harmonic oscillator is well known, it pays to begin with a model with a known analytic result so that we can compare this result with various numerical approximations.
The following simple harmonic oscillator (SHO) models compare different solution techniques. These models are listed in order of complexity.
Review the SHO theory in your Physics 130 (General Physics) or Physics 330 (Intermediate Mechanics) notes and textbooks as appropriate. The mathematics behind the SHO model is covered in Physics 201 (Mathematical Methods for Scientists) and in Math 235 (Differential Equations and Infinite Series).