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### Autonomous Equations Lab

*Note that the system may be slow (latency of <10 seconds) for drawing 33 integral curves for some problems. Unfortunately, I haven't found a way yet to get it to post a "Working" message to let you know it is thinking.*

#### Overview

Autonomous equations are equations of the form $$\frac{dx}{dt}=f(x)$$ where the derivative does not depend on the independent variable, $t$ in this case. Here $x$ is the dependent variable, because it is a function of the independent variable, $t$. Of course, the names of the variables are unimportant. Equations like $$\begin{align} \frac{dy}{dx}&=y^2-1 \\ \frac{dx}{dt}&=\frac{k}{x^2} \\ \frac{dp}{dq}&=.07p(1-p/10000) \end{align} $$ are all autonomous, since the derivative depends only on the dependent variable (the function) and not the independent variable. Autonomous equations are common in practice since the laws of nature are not time-dependent. If we start from $F=mx''$ to create a differential equation, we have position $x$ as the dependent variable and time $t$ as the independent variable. In mechanics problems the force will usually depend on the position of where the particle is, the $x$, but not when it is there, the $t$. So the equations we develop this way will usually be autnomous.Any autonomous equation is naturally separable, so you can find a general solution by integration. But integration can be tricky, and frequently you can use geometric techniques to find answers to practical questions more easily than you can by writing down solutions. In problem 6 of the first lab, you found rules for recognizing when a slope field was autonomous (which in that context we defined as "a function of $y$ alone). We will now build on that geometric understanding to see how you can find certain properties of solutions to autonomous equations, especially behavior as the independent variable tends to $\pm\infty$, from geometric principles without solving the equation. Next week we will use this understanding to optimize harvesting strategies for a renewable resource (e.g. fish).

A few years ago class was cancelled due to snow on the day of this lab. I
made some videos so students could go through the material on their own.
**It is probably best if you don't look at these videos prior to the
lab.** Your lab instructor should answer questions and help you discover
this material in your groups during lab. Ideas you develop on your own are
more likely to stick with you on the test and later than ideas that you
saw in a video. But I've left the videos up becuase (a) they might help
later when you want to review, and (b) once I'd gone to the trouble of
making them, it didn't cost anything to keep them available.

#### Lab Instructions

**Video Introduction**

*Note these video introductions were made using the old java version of the lab.*- Consider the following equations
x' = (x-3)*(x+1) x' = x^2 - 4 x' = x^2 + 1

- Conjecture a relationship between the

curve and the slope field fordx/dt vs. x

based on your results from the first step.x vs. t - Repeat the first step for three more equations of your own choice to test the hypothesis you just created.
- If the results with the next three equations confirmed your hypothesis, wonderful. If not, revise your hypothesis in light of the additional evidence and try again. Repeat until you are satisfied.
**Video Summary**- For an autonomous equation, a solution with a
constant value of the form

is called an equilibrium solution. An equilibrium solution isx(t) = x _{0}*stable*if nearby solutions converge to the equilibrium solution and unstable if nearby solutions move away from the equilibrium solution. For instance, if you draw the 33 integral curves in the first example (

), you should see thatx' = (x-3)*(x+1)

is a stable equilibrium whilex = -1

is an unstable equilibrium. Look through the examples you have created and identify the stable and unstable equilibria.x = 3 - Create a hypothesis to predict what the stable and unstable equilibria
will be for an autonomous differential equation based on the

graph.dx/dt vs. x - As before, test three more equations of your own choice to test the hypothesis you just created.
- If the results with the next three equations confirmed your hypothesis, wonderful. If not, revise your hypothesis in light of the additional evidence and try again. Repeat until you are satisfied.
**Video Finale**

#### Lab Report

Write up a report which includes- A list of all the equations you considered along with sketches of the
graphs (both

anddx/dt vs. x `x vs. t`

) for the equations. - The hypotheses you generated.
- A paragraph for each final hypothesis explaining why the
hypothesis is correct. This explanation should involve the geometric
interpretation of slope and derivative. It is
*not*acceptable just to say the hypothesis works for all the examples. Note that the paragraph should be written in complete sentences using proper grammar and spelling.

If you have any problems with this page, please contact bennett@math.ksu.edu.

©2010 Andrew G. Bennett