# chemical reactions linear system

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:

• Analysis of a single differential equation – either a (1) single first-order linear differential equation
• with constant coefficients OR (2) a single second-order linear differential equation with constant

coefficients.

• Analysis of a system of differential equations – linear, constant coefficient 2-by-2 system of differential
• 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:

• Pond or Lake Pollution
• Home Heating
• Drug delivery/diffusion (Compartmental Model)
• Pesticide in Trees and Soil
• Price-Inventory
• Chemical Reactions
• Any other that fits the above description exactly.
• 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.

pdf

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:

• Spring-mass systems (used in sample project; cannot be chosen for project)
• Any other that fits the above description exactly.
• 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

• Describe the particular solutions of each IVP in language of the application
• (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

• Draw the trajectory of each of the two particular solutions (using Desmos plots)
• Describe the particular solution of each IVP in language of the application
• (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

• Tues. Nov 6th: Your chosen application must be reported to your TA. (Also, you should vote.)
• Tues. Nov. 13th: TAs will address some questions about the project.
• Tues. Nov. 27th: Project is due in recitation.

• A sample project, using the spring-mass system, will be posted. It will include examples of the
• inputs for 3 d (ii) and (iii).

• All work for finding solutions should be put in a neat, hand-written appendix.
• Phase plots may be neatly hand-drawn or copied and pasted from a computer plotting tool.
• Discussions should be typed, and manageable mathematical symbols should be typed. However, long
• or complicated mathematical expressions can be hand-written.

• The responses for many parts listed in the outline should be brief. A sentence or two will suffice
• 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.

• Derivations and realistic values for parameters should be cited.
• For 3 c (i), you may want to use a computational phase portrait creator, like those found at http://
• 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.

• I use the phrase ‘meaningfully different’ in a couple of places. What I mean, for example, is not
• 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.