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# Notes on Diffy Qs - Sage demos for section 2.4

On this page the animations are prerendered. There is no reason to run the Sage code unless you changed it.
**These animations take Sage ***long* time to draw.
If you wish to experiment look for the "srange" in the code to make it into
fewer frames, the syntax is "srange(start,end,step)". In the code given, *the steps are on purpose
bigger than what was used to produce the animations*, if you want to replicate the animations exactly
you have to reduce the step size.
Note that the position of the spring is reversed from where it is drawn in the book, but it does not matter
as the spring is symmetric, and it would seem weird to have the spring be below the mass in the animations below.

## Undamped motion

The graph below is the solution to \(x'' + 4x = 0\) with initial conditions \(x(0) = 0.5\) and \(x'(0) = 1.\)
This represents the undamped motion of a mass on a spring. An animation is drawn of the mass on the spring.
The solution is \(x(t) = 0.5 \cos(2t) + 0.5 \sin(2t).\) To plot something different modify the solution below.

Here is the Sage code to produce this animation:

## Overdamped motion

The graph below is the solution to \(2x'' + 7x' + 3x = 0\) with initial conditions \(x(0) = 2\) and \(x'(0) = -11.\)
This represents overdamped motion of a mass on a spring. The solution is \(x(t) = 4e^{-3t} - 2e^{-0.5t}.\)

Here is the Sage code to produce this animation:

## Underdamped motion

The graph below is the solution to \(16x'' + 8x' + 63x = 0\) with initial conditions \(x(0) = 1\) and \(x'(0) = 1.75.\)
This represents underdamped motion of a mass on a spring. Here we also plot the envelope curves.

Here is the Sage code to produce this animation:

The original code is mainly due to Ryan Burkhart.