Kansas State University

Mathematics Department

Textbook Contents

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Autonomous Equations Lab

Launch 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 autonomous.

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).

Graphing Autonomous Equations

The following link to Launch Autonomous Equations Lab will bring up a new window where you can enter an autonomous differential equation of the form $$ \frac{dx}{dt}=f(x),\qquad x(0)=x_0 $$ and see how it looks in two versions. On the left it will show the graph of $dx/dt$ versus $x$, i.e. the graph of $f(x)$. On the right, it will show the slope field of the differential equation, which is a graph of $x$ vs. $t$. Note that the $x$-axis is therefore horizontal on the left graph and vertical on the right graph. The $x$-axis is marked in green on both axes, and there is a red dot corresponding to the same value of $x$ on both graphs. The dot can be moved on the right graph and the corresponding dot on the left graph will follow. The right graph has buttons to plot either the one integral curve corresponding to the listed initial value or to plot 33 integral curves, just as in the slope field lab. Also note that if you change the equation, you need to click the "Update Graphs" button in the lower right to be sure everything updates.

Identifying Equilibria

  1. Consider the following equations
    • x' = (x-3)*(x+1)
    • x' = x^2 - 4
    • x' = x^2 + 1
    List where x' is positive, negative and 0. List where the slope field for the differential equation is increasing, decreasing, and horizontal.

  2. Conjecture a relationship between the dx/dt vs. x curve and the slope field for x vs. t based on your results from the first step.

    Note that this conjecture need not be correct. It will be judged on whether it is a plausible first guess (though of course it is fine if it is correct).

  3. Repeat the first step for three more equations of your own choice to test the hypothesis you just created. List your three new equations and the same information as you collected in Exercise 1.

  4. 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. List your final hypothesis.

    This time you will be graded on having a correct conclusion.

  5. Write a paragraph explaining why your final 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.

Classifying Equilibria

For an autonomous equation, a solution with a constant value of the form x(t) = x0 is called an equilibrium solution. An equilibrium solution is 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 (x' = (x-3)*(x+1)), you should see that x = -1 is a stable equilibrium while x = 3 is an unstable equilibrium.
  1. Look through the examples you have created and identify the stable and unstable equilibria.

  2. Create a hypothesis to predict what the stable and unstable equilibria will be for an autonomous differential equation based on the dx/dt vs. x graph.

    As before, your initial hypothesis doesn't have to be right (though I think you can probably get it right already).

  3. As before, test three more equations of your own choice to test the hypothesis you just created. List your three new equations and whether your hypothesis correctly predicts the stable and unstable equilibria.

  4. 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. List your final hypothesis.

  5. As in exercise 5, write a paragraph explaining why your final 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.

While you may work with a partner on the lab (indeed, due to space constraints you will pretty much have to work together) everyone should turn in a separate lab report.

If you have any problems with this page, please contact bennett@math.ksu.edu.
©1994-2025 Andrew G. Bennett