Introduction to Ordinary Differential Equations and Some Applications Edward Burkard Introduction This book started out as lecture notes for a class I taught on Ordinary Differential Equations back in the summer of 2010 at the University of California, Riverside. My main motivation was for the students to have a text to use that was not outrageously priced (it was $150 for just a used version of the text!). My general philosophy is that students should not have to go bankrupt buying textbooks, which is why I intend to keep the price of this text as minimal as possible. I also wanted to write a text that just got to the point, while not sacrificing essential details. Many of the students taking this class were not interested in the detailed proofs of all the results, and as such, this text was not written with rigorous proof in mind. I included proofs of easy results and of some essential results, but most of the more involved proofs were either omitted, or left to exercises. Some easy proofs were also left as exercises to help foster the students' proof skills. Given all this, the present text is not appropriate for anything other than a differential equations class that has a class on single variable calculus as an immediate prerequisite. A better text for a more thorough and detailed survey would be [1]. I suppose a brief outline of the book is in order. Chapter 0 begins by covering the immediate prerequisites to be able to take a course in differential equations. Not everything in this chapter is usually covered by the time a typical student takes this class, e.g., Sections 0.4-0.7. In fact, one could rightfully say that Section 0.7 is actually a topic for this class, with which I agree, but I think introducing the topic of a slope field is more appropriate for a prerequisite chapter, then its use is brought in when talking about drawing families of solutions. Chapter 1 covers many different types of first order differential equations, both linear and nonlinear. It ends with a proof of the existence and uniqueness theorem for solutions of first order ODEs, which can be skipped without losing any continuity, but also sacrificing rigor (which is probably ideal in a course targeted at engineers). In Chapter 2, we look at homogeneous and non-homogeneous second order linear differential equations, and various ways of solving them, such as the method of undetermined coefficients and variation of parameters. We introduce a bit of rigor with Section 2.2 where we talk about the Wronskian. We close this chapter with a section on special types of nonlinear second order ODEs, namely those that can be reduced back down to first order ones, and a section on boundary value problems. In Chapter 3, we talk about higher order (≥ 3) linear differential equations, and we generalize the methods of undetermined coefficients and variation of parameters to this case. Chapter 4 covers applications of differential equations from many dif- ferent fields: orthogonal trajectories (math), mixing problems and radioactive decay (chemistry), compound interest (economics), mechanics and E&M (physics/engineering). Finally, in Chapter 5, we cover my favorite topic, that of the Laplace transformation. We cover the transform itself, properties of it, and give an informal treatment of the inverse Laplace transformation (to give a formal treatment, we would need complex analysis, which is beyond the scope of this book). In the latter half of the chapter we use the Laplace transformation to find solutions of some initial value problems, and we use it to solve some Volterra equations, or as they are sometimes referred to as, integro-differential equations. In the next chapter, Chapter 6, we discuss series solutions to differential equations, starting with a discussion of Taylor series. This is necessary since the methods developed earlier in the book might not be sufficient to solve a given equation, and using series is one, rather powerful, way to get around it. The problem here is we have to watch out for singular points x (points that could make the differential equation undefined, e.g., x = 1 in y00 + y0 + y = 2x (we always x − 1 want the highest derivative to have a coefficient of 1)). In Chapter 7, we give an introduction to solving systems of first order linear differential equations by covering 2 × 2 systems of them. We solve them both i ii by force (using calculus only) and by using the more sophisticated method of linear algebra. This chapter ends with an application to the two body problem. In the final chapter we give some ways to get numerical approximations to solutions of differential equations. I would like to thank all of the students in the class for putting up with my notes! I would especially like to thank those who made suggestions or corrections. I would also like to thank my grader for the course, Spencer Shepard, for notifying me of several typos, and for coming in to the office at 11pm to help me finish grading the final for this course. I am sure there are still many more typos, grammar errors, unclear statements, etc. contained in this book, and I warmly welcome you to email me with any suggestions you may have to make the text better! You can email me at [email protected] (at least for now). Writing this text has been a long journey. I started and wrote a majority of it for the class I taught, but there were several more topics I wanted to include, and as the start and end dates show, graduate school had plans for me other than working on this book. Hopefully this book is as useful as I intend it to be. Edward Burkard Begun: April 2010, Riverside, CA Completed: ?? Contents 0 Prerequisites 1 0.1 Precalculus Topics .......................................... 1 0.1.1 Partial Fractions . 1 0.1.2 Trigonometric Identities . 4 0.2 Differentiation Techniques ...................................... 7 0.2.1 The Leibniz Rule . 7 0.2.2 Chain Rule . 7 0.2.3 Implicit Differentiation . 8 0.3 Integration Techniques ........................................ 10 0.3.1 u-Substitution . 10 0.3.2 Integration by Parts . 10 0.3.3 Trigonometric Substitution . 11 0.3.4 Partial Fractions . 12 0.4 Series .................................................. 14 0.4.1 Series . 14 0.4.2 Issues of Convergence . 14 0.4.3 Reindexing . 15 0.4.4 Power Series . 15 0.4.5 Geometric Series . 18 0.4.6 Miscellany . 19 0.5 Complex Numbers and Related Topics .............................. 24 0.5.1 Complex Numbers . 24 0.5.2 Euler's Formula . 25 0.6 Multi-Variable Functions ....................................... 27 0.6.1 Partial Derivatives . 27 0.6.2 Integration of Multivariable Functions . 28 0.7 Miscellaneous Things ......................................... 30 0.7.1 Slope Fields . 30 1 First-Order Differential Equations 35 1.1 Introduction to Differential Equations .......................... 35 1.2 Seperable Differential Equations .............................. 43 1.3 Exact Differential Equations ................................. 48 1.3.1 Exact . 48 1.3.2 Solving Exact Differential Equations . 49 1.3.3 Integrating Factors . 50 1.4 Linear Differential Equations ................................. 55 1.4.1 Variation of Parameters . 55 1.4.2 Integrating Factors . 57 iii iv CONTENTS 1.5 Bernoulli Equations ....................................... 60 1.6 Homogeneous Differential Equations ........................... 62 1.7 Existence and Uniqueness of Solutions to Differential Equations . 65 1.7.1 The Theorem ......................................... 65 1.8 Additional Exercises ....................................... 66 2 Second-Order Differential Equations 71 2.1 Constant Coefficient Homogeneous Linear Equations . 71 2.1.1 The Characteristic Polynomial . 71 2.1.2 Distinct Real Roots . 72 2.1.3 Repeated Real Roots . 73 2.1.4 Complex Roots . 74 2.2 The Wronskian ........................................... 78 2.2.1 General Homogeneous Linear Equations . 78 2.2.2 The Wronskian . 78 2.3 Non-Homogeneous Linear Equations ........................... 82 2.3.1 Superposition . 83 2.3.2 Introduction to Undetermined Coefficients . 84 2.4 Reduction of Order ........................................ 87 2.5 Cauchy-Euler Equations .................................... 91 2.5.1 Indicial Equation . 91 2.5.2 Distinct Real Roots . 91 2.5.3 Repeated Real Roots . 92 2.5.4 Complex Roots . 92 2.6 The Method of Undetermined Coefficients ....................... 95 2.7 Variation of Parameters .................................... 99 2.8 Some Non-Linear Second Order Equations . 103 2.8.1 Missing Dependent Variable . 103 2.8.2 Missing Independent Variable . 104 2.9 Additional Exercises .......................................107 3 Higher-Order Differential Equations 111 3.1 General nth order Linear Differential Equations . 111 3.2 The Method of Undetermined Coefficients . 112 3.3 Variation of Parameters ....................................113 3.4 Additional Exercises .......................................114 4 Some Applications of Differential Equations 115 4.1 Orthogonal Trajectories ....................................115 4.2 Mixing Problems .........................................117 4.3 Modeling Growth and Decay .................................119 4.4 Topics in Mechanics .......................................120 4.4.1 Some Useful Stuff . 120 4.4.2 Simple Spring Motion . 121 4.4.3 Motion Under a Central Force . 122 4.5 Topics in Electricity and Magnetism . 124 4.5.1 The RLC Circuit . 124 4.6 Additional Exercises .......................................126 CONTENTS v 5 Laplace Transformations 127 5.1 The Laplace Transform .....................................127 5.1.1