# Ordinary differential equations

MAT 244Fall 2018# Resources

The notebooks I show on the projector during class are made in Sage, a free math software system based on the Python programming language. You can read them in your web browser, without any extra software, but you won't be able to add your own plots and calculations. If you'd like to run the notebooks yourself, you'll need to install Sage. I'll update the notebooks throughout the semester, so the version you have may not match the currently posted version.

Warm-up

As a warm-up exercise, we used our common sense to reason about what the functions in this notebook could look like. Later, we'll describe the same functions quantitatively using differential equations.

Direction fields

In this notebook, I wrote down differential equations describing the functions in the warm-up. I plotted their direction fields* and some of their solutions.

* In the pendulum scenario, the model doesn't give a direct relationship between θ' and θ, so there's no direction field to plot.

Separation of variables

Here are some separable differential equations, along with their direction fields and solution plots.

On the nature of the function expressive of the law of human mortality

The separable equation we solved as an exercise in class is based on this classic actuarial science paper.

Rescaling the unknown function (integrating factors)

This notebook illustrates how direction fields change when you rescale the unknown function. Sometimes, if you scale just right, you can get a rescaled direction field that only depends on time.

Other rescalings

This notebook illustrates special rescalings of the unknown that you can use to solve (projective) homogeneous equations and Bernoulli equations.

Exact equations

In lecture, we studied the van der Waals temperature function on the volume, pressure plane. This notebook shows the constant-temperature curves, which we described using an exact differential equation.

Approximating solutions

This notebook uses Picard iteration to approximate a function that isn't described by a nice formula, but is described by a nice differential equation.

The Wronskian

These animations show solutions of two second-order differential equations moving in the

*x*,*x*′ plane.Variation of parameters (for second-order linear equations)

The second-order variation of parameters recipe demonstrated in two example problems.

Variation of parameters (for vector equations)

Notes on variation of parameters for vector equations.

Nonlinear systems

Nonlinear systems of equations from ecology and epidemiology.

Experimental Demonstration of Volterra's Periodic Oscillations in the Numbers of Animals

This paper describes an experimental demonstration of the predatory-prey model we studied in class.