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# Notes on Diffy Qs - SAGE demonstrations 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.

## 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 envelop curves.

Here is the SAGE code to produce this animation:

The code is mainly due to Ryan Burkhart, it was then slightly modified by Jiří Lebl.