Ordinary Di erential Equations - Calvin College

1 Ordinary Differential EquationsA linear algebra Perspective(Version )LmgF(t) mg sin F(t) cos Todd KapitulaContentsIntroduction..11 Essentials of linear algebra .. Solving linear systems .. and terminology .. of linear systems .. by Gaussian elimination .. Vector algebra and matrix/vector multiplication .. combinations of vectors .. multiplication .. Matrix algebra : addition, subtraction, and multiplication .. Sets of linear combinations of vectors .. of a set of vectors .. independence of a set of vectors .. independence of a set of functions .. The structure of the solution .. homogeneous solution and the null space .. particular solution .. Equivalence results .. solution exists .. solution always exists .. unique solution exists .. unique solution always exists .. Subspaces .. spaces .. and span .. column space .. Basis and dimension .. and rank .. Inner-products and orthogonal bases.

2 Inner-product onRn.. bases .. bases and Fourier expansions .. Gram-Schmidt procedure .. expansions with trigonometric functions .. The matrix transpose, and two more subspaces .. Subspace relationships .. Least squares .. Matrix algebra : the inverse of a square matrix .. The determinant of a square matrix .. linear algebra with complex-valued numbers, vectors, and matrices .. Eigenvalues and eigenvectors .. Characterization of eigenvalues and eigenvectors .. Properties .. Eigenvectors as a basis, and Fourier expansions .. Case studies .. Voter registration .. Discrete SIR model .. Northern spotted owl .. 118 Group projects .. 123 MATLAB support.. 129 Answers to selected exercises.. 137 References .. 141 Index.. 143 Introductionch:introThis book arose from lecture notes that I began to develop in 2010-2011 for a first course in or-dinary differential Equations (ODEs). At Calvin College the students in this course are primarilyengineers.

3 In our engineering program it is generally the case that the only (formal) linear al-gebra the students see throughout their undergraduate career is what is presented in the ODEcourse. This is not unusual, as the ABET Accreditation Criteria of 2012-13 do not explicitlyrequire a course devoted to the study of linear algebra . Since, in my opinion, the amount ofmaterial on linear algebra covered in, , the classical text of Boyce and DiPrima [10], is in-sufficient if that is all you will see in your academic career, I found it necessary to supplementwith notes on linear algebra of my own design. Eventually, it became clear that in order tohave a seamless transition between the linear algebra and ODEs, there needed to be one is not a new idea; for example, two recent texts which have a substantive linear algebracomponent are by Boelkins et al. [7] and Edwards and Penney [16].Because there is a substantive linear algebra component in this text, I - and more importantly,the students - found it to be much easier later in the text when discussing the solutions of linearsystems of ODEs to focus more on the ODE aspects of the problems, and less on the underlyingalgebraic manipulations.

4 I have found that by doing the linear algebra first, it allowed me tomore extensively and deeply explore linear systems of ODEs. In particular, it is possible to domuch more interesting examples and applications. I believe that this inclusion of more modelingand model analysis is extremely important; indeed, it is precisely what is recommended inthe 2013 report by the National Academy of Sciences on the current state, and future, of themathematical applications presented in this text are labeled Case Studies . I chose this moniker be-cause I wanted to convey to the reader that in solving particular problems we were going todo more than simply find a solution; instead, we were going to take time to determine whatthe solution was telling us about the dynamical behaviour for the given physical system. Thereare 18 case studies presented herein. Some are classical - , damped mass-spring systems,mixing problems (compartment models) - but several are not typically found in a text such asthis.

5 Such examples include a discrete SIR model, a study of the effects on the body of leadingestion, strongly damped systems (which can be recast as a singular perturbation problem),and a (simple) problem in the mathematics of climate. It is (probably) not possible to presentall of these case studies in a one-semester course. On the other hand, the large number allowsthe instructor to choose a subset which will be of particular interest to his/her book is formatted as follows. In Chapter 1 we discuss not only the basics of linear algebrathat will be needed for solving systems of linear Ordinary differential Equations , , Gaussianelimination, matrix algebra , and eigenvalues/eigenvectors, but we discuss such foundationalmaterial as subspaces, dimension, etc. While the latter material is not necessary to solve ODEs,12 IntroductionI find that this is a natural time to introduce students to these more abstract linear algebraconcepts. Moreover, since linear algebra is such foundational material for a mathematical un-derstanding of all of the sciences, I feel that it is essential that the students learn as much asthey reasonably can in the short amount of time that is available.

6 It is typically the case that thematerial in Chapter 1 can be covered in about 15-18 class periods. Primarily because of timeconstraints, when presenting this material I focus primarily on the case of the vector culminating section in the chapter is that on eigenvalues and eigenvectors. Here I espe-cially emphasize the utility of writing a given vector as a linear combination of closing section considers the large-time behavior associated with three discrete dynamicalsystems. If the reader and/or instructor wishes to have a supplementary text for this chapter,the book by Hefferon [23] is an excellent companion. Moreover , the PDF can be had for free the linear algebra has been mastered, we begin the study of ODEs by first solvingscalar first-order linear ODEs in??. We briefly discuss the general existence/uniqueness the-ory, as well as the numerical solution. When solving ODEs numerically, we use the by J. Polking. These MATLAB programs haveaccompanying Java applets: DFIELD: PPLANE: experience is that these software tools are more than sufficient to numerically solve theproblems discussed in this class.

7 We next construct the homogeneous and particular solutionsto the linear problem. In this construction we do three things:(a) derive and write the homogeneous solution formula in such a way that the later notionof a homogeneous solution being thought of as the product of a matrix-valued solutionand a constant vector is a natural extension(b) derive and write the variation-of-parameters solution formula in such a manner that theideas easily generalize to systems(c) develop the technique of undetermined chapter closes with a careful analysis of the one-tank mixing problem under the assump-tion that the incoming concentration varies periodically in time, and a mathematical financeproblem . The idea here is to:(a) show the students that understanding is not achieved with a solution formula; instead, itis necessary that the formula be written correctly so that as much physical informationas possible can be gleaned from it(b) introduce the students to the ideas of amplitude plots and phase plots(c) set the students up for the later analysis of the periodically forced a final note, in many (if not almost all) texts there is typically in this chapter an extensivediscussion on nonlinear ODEs.

8 I chose to provide only a cursory treatment of this topic at theend of this book because of:(a) my desire for my students to understand and focus on linearity and its consequences(b) the fact that we at Calvin College teach a follow-up course on nonlinear dynamics usingthe wonderful text by Strogatz [40].In??we study systems of linear ODEs. We start with five physical examples, three of whichare mathematically equivalent in that they are modeled by a second-order scalar ODE. We showthatnth-order scalar ODEs are equivalent to first-order systems, and thus (hopefully) convincethe student that it is acceptable to skip (for the moment) a direct study of these higher-orderIntroduction3scalar problems. We almost immediately go the case of the homogeneous problem being con-stant coefficient, and derive the homogeneous solution via an expansion in terms of eigenvec-tors. From a pedagogical perspective I find (and my students seem to agree) this to be a naturalway to see how the eigenvalues and eigenvectors of a matrix play a key role in the construc-tion of the homogeneous solution, and in particular how using a particular basis may greatlysimplify a given problem.

9 Moreover, I find that this approach serves as an indirect introduc-tion to the notion of Fourier expansions, which is of course used extensively in a successorcourse on linear partial differential Equations . After we construct the homogeneous solutionswe discuss the associated phase plane. As for the particular solutions we mimic the discussionof the previous chapter and simply show what few modifications must be made in order forthe previous results to be valid for systems. My experience has been that the manner in whichthings were done in the previous chapter helps the student to see that it is not the case we arelearning something entirely new and different, but instead we are just expanding on an alreadyunderstood concept. The chapter closes with a careful analysis of three problems: a two-tankmixing problem in which the incoming concentration into at one of the tanks is assumed tovary periodically in time, a study of the effect of lead ingestion, and an SIR model associatedwith zoonotic (animal-to-human) bacterial infections.

10 As in the previous chapter the goal is tonot only construct the mathematical solution to the problem, but to also understand how thesolution helps us to understand the dynamics of the given physical solve higher-order scalar ODEs. Because all of the theoretical work has already beendone in the previous chapter, it is not necessary to spend too much time on this particular particular, there is a relatively short presentation as to how one can use the systems theoryto solve the scalar problem. The variation of parameters formula is not re-derived; instead, it isjust presented as a special case of the formula for systems. We conclude with a careful study ofseveral problems: the undamped and damped mass-spring systems, a ( linear ) pendulum drivenby a constant torque, a couple mass-spring system, and the vibrations of a beam. The last studyintroduces the separation of variables technique for solving linear PDEs. Nice illustrative Javaapplets for the mass-spring problems are: Forced and damped oscillations of a spring pendulum: Coupled oscillators: are also illustrative movies which are generated by solve scalar ODEs using the Laplace transform.