[rodh]

Would second order differentials imply linear acceleration?

Would we then always get better approximations the higher the order?

EDIT: I ask this because linear acceleration doesn't seem like something which might readily occur in nature.

[dietrichepp]

You can get constant (not linear) acceleration with a second order differential equation. Here is an example:

    d^2 x
    ----- = -g
    dt^2

Solving this gives us the parabola we are all familiar with, when an object falls under constant acceleration:

    x = -1/2 g t^2 + v_0 t x + x_0

This is a second order differential equation, but it is too trivial an example. Here is a more interesting differential equation:

    d^2 x
    ----- = -x
    dt^2

All solutions to this equation take the form:

    x = a sin t + b cos t

There are a lot of physical materials in nature which produce this kind of relationship between force and position. Basically, anything that is kind of like a spring. Reach over and flick a glass on your desk with your fingernail. Hear it ring? The sine wave that it's ringing with is predicted by the relationship between force and position that we understand. Every tuned musical instrument in the world, with one exception, generates notes that can be modeled by some differential equation. Pianos, violins, and horns are relatively easy; drums are much harder but still tractable.

The chemical and physical systems in living tissue are much more complicated. Our best models for them often incorporate nonlinear elements (the sine wave example is linear) or a large number of variables. But periodic behavior is still quite common, and the sine wave is in some senses the simplest function that is smooth and periodic.

Note: The one musical instrument for which differential equations are not the best model is the digital synthesizer.