For the project this semester, you will choose a physical, biological, economic, or social system that can

be modeled by a linear, constant coefficient 2-by-2 system of differential equations. Your project will have

two main pieces:

with constant coefficients OR (2) a single second-order linear differential equation with constant

coefficients.

equations.

Most applications are of the first type: when one unknown quantity is considered, the model is a single

first-order linear differential equation with constant coefficients; when that quantity interacts with another

quantity, the model is a 2-by-2 linear, constant coefficient 2-by-2 system of differential equations. Examples

include:

You can use this chapter of a differential equations text by G.B. Gustafson from the University of Utah to

get descriptions and details of some of these systems: www.math.utah.edu/~gustafso/2250systems-de.

Some applications are of the second type: the model is a single second-order linear differential equation

with constant coefficients. It can be analyzed as such with the methods of Lebl Chapter 2. However, the

second-order equation can be converted to a 2-by-2 system and analyzed with the methods of Lebl Chapter

3. Examples include:

1. Outline of Project Write-up

(1) Introduction

(2) Single Equation

(a) Derivation of equation and meaning of parameters (include units)

(b) Meaning and Relevance of Homogeneous vs Non-Homogenous

(c) General Solution

(d) An Initial Value Problem and a Particular Solution (Typically Non-Homogeneous)

(e) Behavior: Discuss solution in the language of the application

(3) System of ODEs

(a) Derivation of equation and meaning of parameters (including units)

(b) Meaning and Relevance of Homogeneous vs Non-Homogenous

(c) Homogenous System

(i) Pick two sets of values for parameters that will give two of the three possibilities for

eigenvalues: (1) distinct, real; (2) complex conjugate pair; (3) repeated (or multiple)

eigenvalues (I suggest using computational aids for this.)

(ii) For each of the two situations:

(A) Discuss how realistic the parameter values are. (At least one set of values should berealistic.)

(B) Give general solution

(C) Draw a phase portrait

(D) Give two (meaningfully different) initial conditions and their particular solutions

(d) Non-homogeneous

(i) Meaning of the inhomogeneity

(ii) Find the general solution using Wolframalpha or another symbolic computational aid.

(iii) Plot the general solution on Desmos.com

(iv) Give two situations (by changing initial conditions or the forcing) that lead to meaning-

fully different behaviors

(4) Appendix

(a) Hand-written work for finding general solution of single equation

(b) Hand-written work for finding general solutions of 2-by-2 systems

2. Dates

3. Comments

inputs for 3 d (ii) and (iii).

or complicated mathematical expressions can be hand-written.

for many pieces. Derivations in 2 (a) and 3 (a) should be a short paragraph. Descriptions of the

behavior of solutions in the language of your application (2(e), 3 c (ii) (D), 3 d (iv)) should be your

longest written sections, but still do not need to be longer than 4-5 sentence paragraphs.

mathlets.org/mathlets/linear-phase-portraits-matrix-entry/ and http://parasolarchives.

com/tools/phaseportrait. You can try different parameters and see the phase portraits associated

with them to choose your two sets of parameters.

to choose initial conditions (0, 1) and (0, 1.1). The behaviors will be nearly identical. Choose (0, 1)

and (−1, 0) or (1, 10) and (10, 1). It will depend on your system, but there should be something

interesting to say about the behavior for the two ‘meaningully different’ choices.