Here is an example, if you know calculus and introductory physics.
1) Taken Newton's law F = ma and write it as F = m x''(t), x(t) is a particles position and so x'(t) is its velocity and x''(t) is acceleration.
2) Consider the following assumption (Hooke): the force exerted by a spring is proportional to its extension, and opposite in direction to that extension. In equation form this says F =  k x, where k is a constant of proportionality, x is the extension from equilibrium, and the minus sign insures that the force is opposite the extension.
3) Newtons second law now becomes m x''(t) = k x(t). Take the case k = m = 1 and this reduces to x'' = x, or "the second derivative is the negative of the original function". Functions that have this property are the sine and cosine, which upon reflection seem to describe the oscillation of a spring quite well.
