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Transcript of ORDINARY DIFFERENTIAL EQUATIONS - .1.1 Basic Concepts and Terminology 3 The first five equations

  • William A. Adkins, Mark G. Davidson

    ORDINARY DIFFERENTIAL

    EQUATIONS

    August 20, 2007

  • Contents

    1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Basic Concepts and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Direction Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    2 First Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 372.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Linear First Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.3 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.4 Miscellaneous Nonlinear First Order Equations . . . . . . . . . . . . . . 70

    3 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.1 Definition and Basic Formulas for the Laplace Transform . . . . . 803.2 Partial Fractions: A Recursive Method for Linear Terms . . . . . . 953.3 Partial Fractions: A Recursive Method for Irreducible

    Quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103.4 Laplace Inversion and Exponential Polynomials . . . . . . . . . . . . . 1153.5 Laplace Inversion involving Irreducible Quadratics . . . . . . . . . . . 1203.6 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1303.7 Laplace Transform Correspondences . . . . . . . . . . . . . . . . . . . . . . . 1383.8 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.9 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583.10 Table of Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

    4 Linear Constant Coefficient Differential Equations . . . . . . . . . 1634.1 Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1634.2 The Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . 1714.3 Linear Homogeneous Differential Equations . . . . . . . . . . . . . . . . . 1794.4 The Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . 1844.5 The Incomplete Partial Fraction Method . . . . . . . . . . . . . . . . . . . 192

  • VIII Contents

    5 System Modeling and Applications . . . . . . . . . . . . . . . . . . . . . . . 1975.1 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1975.2 Spring Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2135.3 Electrical Circuit Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

    6 Second Order Linear Differential Equations . . . . . . . . . . . . . . . . 2276.1 The Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . 2286.2 The Homogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2336.3 The Cauchy-Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2386.4 Laplace Transform Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2416.5 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2516.6 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

    7 Power Series Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2617.1 A Review of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2617.2 Power Series Solutions about an Ordinary Point . . . . . . . . . . . . . 2747.3 Orthogonal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2867.4 Regular Singular Points and the Frobenius Method . . . . . . . . . . 2907.5 Laplace Inversion involving Irreducible Quadratics . . . . . . . . . . . 319

    8 Laplace Transform II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3318.1 Calculus of Discontinuous Functions . . . . . . . . . . . . . . . . . . . . . . . 3328.2 The Heaviside class H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3468.3 The Inversion of the Laplace Transform . . . . . . . . . . . . . . . . . . . . 3548.4 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . 3588.5 The Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3638.6 Impulse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3698.7 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3728.8 Undamped Motion with Periodic Input . . . . . . . . . . . . . . . . . . . . . 3848.9 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

    9 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3999.1 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3999.2 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4089.3 Invertible Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4259.4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

    10 Systems of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 44110.1 Systems of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 441

    10.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44110.1.2 Examples of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . 446

    10.2 Linear Systems of Differential Equations . . . . . . . . . . . . . . . . . . . . 45210.3 Linear Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 46610.4 Constant Coefficient Homogeneous Systems . . . . . . . . . . . . . . . . 47810.5 Computing eAt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48810.6 Nonhomogeneous Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 497

  • Contents IX

    A Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505A.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

    B Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

    C Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

    Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

  • List of Tables

    3.1 Basic Laplace Transform Formulas . . . . . . . . . . . . . . . . . . . . 943.9 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583.10 Table of Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

    C.1 Laplace Transform Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 581C.2 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . 582C.2 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . 583C.2 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . 584C.3 Table of Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584C.3 Table of Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

  • 1

    Introduction

    1.1 Basic Concepts and Terminology

    What is a differential equation?

    Problems of science and engineering frequently require the description of someproperty of interest (position, temperature, population, concentration, cur-rent, etc.) as a function of time. However, it turns out that the scientific lawsgoverning such properties are very frequently expressed as equations relatinghow the various rates of change of the property are related to the quantityat a particular time. For example, Newtons second law of motion states (inwords):

    Force = MassAcceleration.For the simple case of a particle of mass m moving along a straight line, if welet y(t) denote the distance of the particle from the origin, then accelerationis the second derivative of position, and hence Newtons law becomes themathematical equation

    F (t, y(t), y(t)) = my(t), (1)

    where the function F (t, y, y) gives the force on the particle at time t, a dis-tance y(t) from the origin, and velocity y(t). Equation (1) is not an equationfor y(t) itself, but rather an equation relating the second derivative y(t), thefirst derivative y(t), the position y(t), and t. Since the quantity of interest isy(t), it is necessary to be able to solve this equation for y(t). Our goal in thistext is to learn techniques for the solution of equations such as (1).

    In the language of mathematics, laws of nature such as Newtons secondlaw of motion are expressed by means of differential equations. An ordinarydifferential equation is an equation relating an unknown function y(t), someof the derivatives of y(t), and the variable t, which in many applied problemswill represent time. Like (1), a typical ordinary differential equation involvingt, y(t), y(t), and y(t) would be an equation

  • 2 1 Introduction

    (t, y(t), y(t), y(t)) = 0. (2)

    In such an equation, the variable t is frequently referred to as the indepen-dent variable, while y is referred to as the dependent variable, indicatingt