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Differential Equations with Laplace 39 Washington State University MATH 315 Fall 2009 Lecture Notes Differential Equations Authors: Travis Mallett | Josh Fetbrandt Contents 1 Introduction to Differential Equations8 1.1 What This Means to You......................9 2 First-Order Equations 10 2.1 Solutions to Separable Equations.................. 10 2.1.1 Summary of Separable Differential Equations....... 11 2.2 Linear Equations and the Integrating Factor............ 12 2.2.1 Summary of Integrating Factor Technique......... 13 2.3 Mixing Problems........................... 14 2.4 Exact Differential Equations..................... 16 2.5 Autonomous Equations and Stability................ 17 2.5.1 Example of Qualitative Analysis.............. 18 3 Second-Order Equations 20 3.1 General Solutions, Linearity, and Other Hoopla.......... 20 3.1.1 Homogeneous Equations................... 22 3.1.2 Inhomogeneous Equations.................. 23 3.2 Techniques for Solving Second-Order Equations.......... 25 3.2.1 The Characteristic Equation................ 25 3.2.2 Method of Undetermined Coefficients........... 26 3.2.3 Variation of Parameters................... 29 3.3 Harmonic Motion (avoid this section)............... 29 3.3.1 Lots of Words......................... 29 3.3.2 Four Types of Harmonic Motion.............. 31 3.3.3 Stable and Unstable Motion (Resonance)......... 32 3.3.4 Summary of Harmonic Motion and Why you Shouldn't Care.............................. 33 3.3.5 Awesome Table That May Not Help............ 34 4 Laplace Transform 35 4.1 Intuition, Gut-Feelings, and What this Means........... 35 4.2 Groaning Forward: Examples of Laplace Transforms....... 36 4.3 Inverse Laplace Transform...................... 37 4.4 Solving Differential Equations with Laplace............ 39 3 4 CONTENTS 4.5 The Heaviside Function (Unit Step)................ 41 4.5.1 Laplace Transform of the Heaviside............. 42 4.5.2 Inverse Laplace Transform of the Heaviside (and the shift idea).............................. 43 4.6 Dirac Delta Function......................... 45 4.6.1 Laplace Transforms of the Delta Function......... 46 4.7 Discontinuous Forcing Terms.................... 47 5 Linear Algebra Review 48 5.1 Systems of Equations, the Matrix, and Solutions......... 48 5.1.1 No Solution Exists...................... 49 5.1.2 One and Only One Solution................. 49 5.1.3 Infinitely Many Solutions.................. 50 5.2 Eigenvalues and Eigenvectors.................... 51 6 Introduction to Systems 54 7 Solving Systems of Differential Equations 55 7.1 Graphing Solutions of Systems................... 55 7.1.1 Saddle Solutions....................... 57 7.1.2 Nodal Sink.......................... 58 7.1.3 Nodal Source......................... 59 7.1.4 Spiral Sink.......................... 60 7.1.5 Spiral Source......................... 61 8 Series Solutions to Differential Equations 62 8.1 Ummm...What Are They?...................... 62 8.2 Solving Differential Equations Using Series Solutions....... 63 A Practice Exams 66 A.1 Exam Information.......................... 66 A.2 Sample Test 1............................. 68 A.3 Sample Test 2............................. 69 A.4 Test 1 With Answers......................... 70 A.5 Test 2 With Answers......................... 76 A.6 Test 3 With Answers......................... 83 B Resources 92 B.1 Video Lectures............................ 92 B.2 Books................................. 93 B.3 Online Resources........................... 93 C Table of Laplace Transforms 94 List of Figures 2.1 Mixing Problems........................... 15 2.2 Qualitative Analysis for f(y) = 0.................. 19 2.3 Qualitative Analysis of Solution................... 19 3.1 How to Solve Homogeneous Equations............... 23 3.2 How to Solve Inhomogeneous Equations.............. 24 3.3 Table of Harmonic Motion...................... 34 4.1 Heaviside Function H(t)....................... 41 4.2 Shifted Heaviside Function H(t − a)................ 41 4.3 Subtracting Heaviside Functions to Get a \Pulse"......... 41 4.4 \Masking" Graphs with Heaviside................. 42 4.5 Dirac Delta Function......................... 46 5.1 No Solution as Parallel Lines.................... 49 5.2 One and Only One Solution..................... 50 7.1 A Generic Spiral (Don't look at this too long!).......... 56 7.2 Saddle Solution............................ 57 7.3 Nodal Sink Solution......................... 58 7.4 Nodal Source Solution........................ 59 7.5 Spiral Sink Solution......................... 60 7.6 Spiral Source Solution........................ 61 A.1 How to Ace the Exams........................ 67 A.2 Scary, Multi-dimensional Mixing Problem............. 85 A.3 A Random Phase Portrait...................... 89 5 List of Tables 2.1 Given Information in Mixing Problems............... 15 3.1 \Method" of Undetermined Coefficients.............. 27 3.2 Spring-Mass Constants for Harmonic Motion Equations..... 30 3.3 Specific Spring-Mass Constants................... 31 4.1 Short Table of Laplace Transforms................. 37 4.2 Even Shorter Table of Laplace Transforms............. 43 6 Dear Reader These notes were written by two students (Travis Mallett and Josh Fetbrandt) at Washington State University in Fall 2009 and are intended to be used with Differential Equations with Boundary Value Problems (Second Edition) by John C. Polking. As we were studying differential equations using this book, we be- gan to find it extremely dysfunctional in many respects. Its poor explanations and confusing structure made it very difficult to learn anything. In the end, we decided to write our own material (based on online lectures and other books) to make up for Polking's book. Although these notes were a direct response to this specific textbook, our work should be applicable to any standard differential equations class. We have abandoned many of the formal aspects of the traditional textbook but all of the information in these notes should be technically correct without the bore of mathematical formalities. We have also tried to obtain a good balance between the practical and theoretical. While the authors of our textbook waved their hands a lot and did not explain concepts, we have tried to rectify this by telling you not only how to use the equations but give explanations about why they work where appropriate. It was our intention to relate many of the concepts to things that should be intuitive to the reader. We feel that it is important to understand (from a gut-feeling level) what is going on. Otherwise many of these concepts drift into obscurity in our minds if they have no practical ramifications. Whether we actually accomplished all of these goals in our notes is another matter entirely. But we did our best under the circumstances (being in school and all). The notes should also be fun to read and we hope you enjoy them. We incorpo- rated many wise-cracks and jokes into the material for entertainment. Although the humor really has no redeeming value, we hope it at least puts a smile on your face a few times to make up for the fact that you are doing differential equations of all things. We have certainly learned a lot from writing these notes and hope you find them useful as you embark on your study of differential equations. If you have any questions or comments, please feel free to look us up. Sincerely, Travis Mallett and Josh Fetbrandt 7 Chapter 1 Introduction to Differential Equations Chances are, that in you journey through mathematics you have heard the term “Differential Equation." You may even vaguely remember something called a \Separable Differential Equation" from your highschool or college first semester calculus courses. If you don't know what a differential equation is, then this section is for you. The world we live in contains many interrelated, changing entities. The earth moves with time as it orbits around the sun, tides are a result of tidal forces created by the moon and attributes of a gas such as pressure, volume, and temperature are all interrelated and dependent on one another. These and many other aspects of science speak of changing quantities. We call changing quantities variables and their rate of change their derivatives. Often we will want to express some relationship between a variable and its derivative to model some physical system or another. For example, we might want to express the speed of a roller coaster in terms of its position. We could somehow mathematically determine that the speed of a (very boring) roller coaster might decrease by some amount for every 100 feet it traveled: dx x = −k dt 100 And this, my friends, is what we call an Ordinary, Separable Differential Equa- tion! We'll get to the ordinary and separable part later. For now, it is enough to know that whenever we write an equation involving a variable and its deriva- tive, we call it a differential equation (for reasons that are obvious enough). In the above equation we see the speed of the roller coaster (the derivative) is some function of the position (designated by the usual x variable) with a weird k fac- tor floating around in there which will tell us how much the speed is decreasing (and some other things). Because we have both the variable x and its derivative 8 1.1. WHAT THIS MEANS TO YOU 9 dx dt in the same equation, it is a differential equation. Another example of a differential equation in nature is Newton's Law of Cooling. This is a model that describes, mathematically, the change in temperature of an object in a given environment. The law states that the rate of change (in time) of
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