MATH0039 Differential & Integral Calculus Project 2
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MATH0039 Differential & Integral Calculus
Project 2: Differential Equations
Due Date: Week 10
Please note, you must include an introduction, methodology, and conclusion in order to receive full credit.
Project Instructions: The aim of the mini projects is to explore the applicability of the mathematical techniques learned in class. In this project, you will tackle a problem using an analytical and numerical method. Your results should be typed up as a report/academic paper and should be a maximum of 3 pages long. This is to encourage conciseness in your writing and presentation of your solutions. Your fellow MATH0039 classmates are your intended audience, so be sure to avoid skipping steps in your calculations. The report should include the following:
1. Introduction: What is the problem we are trying to solve? Is there any background context needed for the reader?
2. Methods: This section will include two subsections: the analytical and numerical solu- tion.
● Analytical solution: Solve the problem using the differential equation techniques we’ve learned in class. Be sure to include all steps of the calculation.
● Numerical solution: Solve the problem using a differential numerical equation technique, e.g. trapezium method. You are welcome to other numerical techniques not learned in class, if you so desire.
● Conclusion: What was the final answer to the original problem? Describe the context of your final answer. For example, if we calculated the circumference of the Earth and found it to a 1cm accuracy, we might say that this level of accuracy is unnecessary and round up our final answer to a more fitting unit like kilometers. What were the answers for each method and how do they compare? What are the benefits and limitations of each method? Which method was more appropriate to this solution? How might you improve the numerical solution?
Options for Project 2: Choose only one question to answer for your project:
1. A sealed tank of bio-nutrient contains two different types of organism that interact with each other. The rates of change of each organism can be modelled using the differential
equations
dz
= -3z + 2y
dt
where z is the number of organism M and y is the number of organism N at time t days. Initially z = 10 and y = 20. How many organisms are there at t = 3 days, and what happens to the populations at t = o?
2. The Harrod-Domar model was original created by Roy Harrod and Evsey Domar (Harrod, 1939). Their model can be used to explain an economy’s growth rate in terms of savings and capital. The model is given by
1 dy
= sc - 6
where s is the savings rate, c is the capital productivity, 6 is the capital depreciation rate, and y is the income and is a function of K, the capital stock (in other words f(K) = y). What is the income at t = 500 if s = 30%, c = 2K5, and 6 = $2? What happens to the economy’s growth rate if the initial condition is f(0) = y6 = 0?
References
R. F. Harrod. An essay in dynamic theory. The Economic Journal, 49(193):14–33, 1939. ISSN 00130133, 14680297. URL http://www .jstor .org/stable/2225181.
2022-12-03
Differential Equations