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
• 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. pdfSome 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)• LRC Circuits
• 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 foreigenvalues: (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 be realistic.)(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
(a) Hand-written work for finding general solution of single equation (b) Hand-written work for finding general solutions of 2-by-2 systems
- Tues. Nov. 27th: Project is due in recitation.3. Comments
- 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, longor complicated mathematical expressions can be hand-written.
- The responses for many parts listed in the outline should be brief. A sentence or two will sufficefor 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-portrait… 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.