Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 1 - Introduction

Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 1 - Introduction

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CHAPTER 1 Introduction T he mathematical sub-discipline of differential equations and dynamical systems is foundational in the study of applied mathematics. Differential equations arise in a variety of contexts, some purely theoretical and some of practical interest. As you read this textbook, you will Fnd that the qualitative and quantitative study of differential equations incorporates an elegant blend of linear algebra and advanced calculus. ±or this reason, it is expected that the reader has already completed courses in (i) linear algebra; (ii) multivariable calculus; and (iii) introductory differential equations. ±amiliarity with the following topics is especially desirable: ± ±rom basic differential equations: separable differential equations and separa- tion of variables; and solving linear, constant-coefFcient differential equations using characteristic equations. ± ±rom linear algebra: solving systems of m algebraic equations with n un- knowns; matrix inversion; linear independence; and eigenvalues/eigenvectors. ± ±rom multivariable calculus: parametrized curves; partial derivatives and gradients; and approximating a surface using a tangent plane. Some of these topics will be reviewed as we encounter them later—in this chapter, we will recall a few basic notions from an introductory course in differential equations. Readers are encouraged to supplement this book with the excellent textbooks of Hubbard and West [ 5 ], Meiss [ 7 ], Perko [ 8 ], Strauss [ 10 ], and Strogatz [ 11 ]. Question: Why study differential equations? 1
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2 Answer: When scientists attempt to mathematically model various natural phenomena, they often invoke physical “laws” or biological “principles” which govern the rates of change of certain quantities of interest. Hence, the equations in mathematical models tend to include derivatives. For example, suppose that a hot cup of coffee is placed in a room of constant ambient temperature α . Newton’s Law of Cooling states that the rate of change of the coffee temperature T ( t ) is proportional to the difference between the coffee’s temperature and the room temperature. Mathematically, this can be expressed as d T d t = k ( T - α ) , where k is a proportionality constant. Solution techniques for differential equations ( de s) depend in part upon how many independent variables and dependent variables the system has.
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This note was uploaded on 02/27/2012 for the course MATH 532 taught by Professor Reynolds during the Fall '11 term at VCU.

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Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 1 - Introduction

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