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Unformatted text preview: Chapter 8: Mathematical Modeling with Differential Equations Summary: This chapter brings together the two important ideas of differentia- tion and integration of functions. These ideas are related by looking at equations that involve both a function, y ( x ) and some of its derivatives, (i.e., y ′ ( x ) , y ′′ ( x ) , etc.) which may be solved in some cases using integration. Topics discussed in this chapter include how to solve first order differential equations, how to use differential equations to model various physical situations, and how to find the unique solution of a differential equation that is paired with an initial condition. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Determine the order of a differential equation (§8 . 1). 2. Determine if a given function is a solution of a differential equation (§8 . 1). 3. Solve a first order separable differential equation (§8 . 2). 4. Draw a slope field or direction field for a differential equation (§8 . 3). 5. Use Euler’s method to approximate the solution of a differential equation (§8 . 3). 6. Solve a first order linear differential equation (§8 . 4). 7. Model various applications using first order differential equations (§8 . 1, §8 . 2, and §8 . 4). 8.1 Modeling With Differential Equations PURPOSE: To define what a differential equation is and to solve some first order differential equations. 165 166 This section introduces what differential equations are and what it means for a function to solve a differential equation. Two types of differential equations are discussed: first order linear equations and separable equations. CAUTION: Differential equations will involve some unknown function y = y ( t ) . Usually this function is just written as y although it is still a function of some independent variable. A first-order differential equation is an equation that involves only y , its derivative, y ′ and an independent variable t . Sometimes, the independent variable will not occur explicitly in the equation. For example, y ′ = y 2 , really could be written as y ′ ( t ) = y ( t ) 2 . A function, y = φ ( t ) , is a solution of the differential equation if an equality results solution general solution (i.e., both sides of the equation are equal) when the function is inserted into the differential equation. A general solution of a differential equation involves a constant of integration that allows for an infinite number of different functions to solve the differential equation. An Initial-Value Problem (IVP) is a differential initial-value problem (IVP) equation that is paired with some initial condition. A solution of an IVP must solve the differential equation and also satisfy the initial condition(s). Initial conditions initial conditions are substituted into the general solution in order to determine a unique value for the constant of integration....
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This note was uploaded on 06/01/2010 for the course CAL 1000 taught by Professor Lee during the Spring '10 term at École Normale Supérieure.
- Spring '10