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Unformatted text preview: Chapter 13 More Differential Equations 13.1 Introduction In our discussion of exponential functions, we briefly encountered the idea of a differential equation. We saw that verbal descriptions of the rate of change of a process (for example, the growth of a population) can sometimes be expressed in the format of a differential equation, and that the functions associated with such equations allow us to predict the behaviour of the process over time. In this chapter, we will develop some of these ideas further, and collect a variety of methods for understanding what differential equations mean, how they can be understood, and how they predict interesting behaviour of a variety of physical and biological systems. First, a brief review of what we have seen about differential equations so far: 1. A differential equation is a statement linking the rate of change of some state variable with current values of that variable. An example is the simplest population growth model: If N ( t ) is population size at time t : dN dt = kN. 2. A solution to a differential equation is a function that satisfies the equation. For instance, the function N ( t ) = Ce kt (for any constant C ) is a solution to the above unlimited growth model. (We checked this by the appropriate differentiation in a previous chapter.) Graphs of such solutions (e.g. N versus t) are called solution curves. 3. To select a specific solution, more information is needed: Namely, some starting value (initial condition) is needed. Given this information, e.g. N (0) = N , we can fully characterize the desired solution. 4. So far, we have seen simple differential equations with simple functions for their solutions. In general, it may be quite challenging to make the connection between the differential equation (stemming from some application or model) with the solution (which we want in order to understand and predict the behaviour of the system.) In this chapter we will expand our familiarity with differential equations and assemble a variety of techniques for understanding these. We will encounter both qualitative and quantitative methods. Geometric as well as algebraic techniques will form the core of the concepts here discussed. v.2005.1 - September 4, 2009 1 Math 102 Notes Chapter 13 13.2 Review and simple examples 13.2.1 Simple exponential growth and decay Consider the simplest differential equation representing an exponential growth model dy dt = y with initial condition y (0) = y . We know that solutions are y ( t ) = y e t . These are functions that grow with time, as shown on the left panel in Figure 13.1. On the other hand, the differential equation dy dt =- y with initial condition y (0) = y has solutions of the form y ( t ) = y e t , which are functions that decrease with time. We show some of these on the right panel of Figure 13.1....
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This note was uploaded on 05/19/2011 for the course MATH 102 Math 102 taught by Professor Allard during the Winter '09 term at The University of British Columbia.
- Winter '09