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) • 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 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 be realistic.) (B) Give general solution 1
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(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 meaningfully 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. 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, 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.