# Writing a mathematical project within certain topics

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For the project this semester, you will choose a physical system that can be modeled by a linear, constant coefficient 2-by-2 system of differential equations. Your project will have the following main pieces: • Analysis of two homogeneous linear system of two first-order differential equations. • Analysis of two non-homogeneous linear system of two first-order differential equations. Application/Topic Choices • Pond or Lake Pollution • Home Heating • Drug delivery/diffusion. • Pesticide in Trees and Soil • Chemical Reactions • Competing Species • LRC Circuits 1. Outline of Project Write-up (1) Introduction (2) System of ODEs (a) Give the most general form of the system of ODEs that models your application. (b) Describe the Meaning and Relevance of an Inhomogeneity (Forcing Function) (c) Homogeneous System (i) Use the two sets of values for parameters given on the Project Assignment page on myCourses. (ii) For the first set of parameters: (A) Discuss the physical meaning of the parameters, including units, and the relative size of the parameters. (B) Give general solution (computed by hand; computations attached as an appendix). (C) Sketch a phase portrait by making use of a computational aid that will take the system itself as input and produce a vector field and trajectories of solutions. (https:// homepages.bluffton.edu/~nesterd/java/slopefields.html will work for many systems. You can click on the plane to have trajectories plotted.) (D) Discuss which trajectories are physically possible and describe what happens to trajectories after long times. (Do they have to be in a certain quadrant?) (E) Give an interesting initial condition, which creates an IVP. • Plot the trajectory of the particular solution to this IVP using Desmos.com • Describe the particular solution of the IVP in language of the application. (iii) For the second set of parameters: Repeat steps (A)-(E) above. (d) Non-homogeneous (i) Give an example of a non-zero constant (vector-valued) forcing function and a (vectorvalued) forcing function that has at least one component that is a sinusoidal function shifted so that it is always positive. (Include units.) (ii) Describe the physical meaning or interpretation of these forcing functions. (iii) For constant forcing (A) Write the non-homogeneous system of ODEs, using the first set of parameters for your topic. (B) Find the general solution, using Wolframalpha or another symbolic computational aid to find a particular solution. (C) Draw a trajectory of a particular solution using Desmos. (Just choose some values for the constants.) (D) From the trajectory, determine the initial condition. (E) Describe the behavior of the particular solution of the IVP in language of the application. (iv) For the (shifted) sinusoidal forcing: Repeat steps (A)-(E) above. (e) Non-homogeneous with Discontinuous Forcing 1 2 (i) Give an example of a discontinuous (vector-valued) forcing function. (ii) Describe the physical meaning or interpretation of this forcing function. (iii) (Extra Credit: 5 pts) Perform the Laplace Transforms of the equations in your IVP with this forcing function and the same paramters and initial condition as above. Discuss how the solution to the IVP could be found, without doing the rest of the computations. (iv) (Extra Credit: 10 pts) Use Laplace Transforms to solve this system. (3) Appendix (a) Hand-written work for finding general solutions of homogeneous systems 2. Dates • Thur. Nov 14th: The initial set up of the systems of ODEs for you application due in the homework assignment. • Thurs Nov 21st: Some plotting of phase portraits and trajectories for your application due in the homework assignment. • Thurs Dec 5th: Project is due in recitation. 3. Comments • A sample project, using the spring-mass system, will be posted. • All work for finding solutions should be put in a neat, hand-written appendix. • Phase plots and trajectories 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) should be a short paragraph. Descriptions of the behavior of solutions in the language of your application should be your longest written sections, but they do not need to be longer than 4-5 sentence paragraphs. • If you use a reference (for the derivation, for instance), it should be cited. • Some examples of entries into online computational and graphing systems will be posted.

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