18.03 Supplementary NotesSpring 2012c Haynes R. Miller and MIT, 2004, 2006, 2008, 2010, 2012Contents0. Preface 11. Notation and language 32. Modeling by first order linear ODEs 63. Solutions of first order linear ODEs 104. Sinusoidal solutions 165. The algebra of complex numbers 236. The complex exponential 277. Beats 348. RLC circuits 389. Normalization of solutions 4110. Operators and the exponential response formula 4511. Undetermined coefficients 5312. Resonance 5513. Time invariance 5814. The exponential shift law 6015. Natural frequency and damping ratio 6316. Frequency response 6617. The Tacoma Narrows bridge 7618. Linearization: the phugoid equation as example 7919. The Wronskian 8720. More on Fourier series 9021. Steps, impulses, and generalized functions 10122. Generalized functions and differential equations 10723. Impulse and step responses 10924. Convolution 11225. Laplace transform technique: coverup 11526. The Laplace transform and generalized functions 11927. The pole diagram and the Laplace transform 12428. Amplitude response and the pole diagram 13029. The Laplace transform and more general systems 13230. First order systems and second order equations 13431. Phase portraits in two dimensions 1380. PrefaceThis packet collects notes I have produced while teaching 18.03, Or-dinary Differential Equations, at MIT in 1996, 1999, 2002, 2004, 2006,2008, 2010, and 2012. They are intended to serve several rather differ-ent purposes, supplementing but not replacing the course textbook.In part they try to increase the focus of the course on topics andperspectives which will be found useful by engineering students, whilemaintaining a level of abstraction, or breadth of perspective, sufficientto bring into play the added value that a mathematical treatment offers.For example, in this course we use complex numbers, and in partic-ular the complex exponential function, more intensively than Edwardsand Penney do, and several of the sections discuss aspects of them.This ties in with the “Exponential Response Formula,” which seems tome to be a linchpin for the course. It leads directly to an understandingof amplitude and phase response curves. It has a beautiful extensioncovering the phenomenon of resonance. It links the elementary theoryof linear differential equations with the use of Fourier series to studyLTI system responses to periodic signals, and to the weight functionappearing in Laplace transform techniques. It allows a direct pathto the solution of sinusoidally driven LTI equations which are oftensolved by a form of undetermined coefficients, and to the expression ofthe sinusoidal solution in terms of gain and phase lag, more useful andenlightening than the expression as a linear combination of sines andcosines.As a second example, I feel that the standard treatments of Laplacetransform in ODE textbooks are wrong to sacrifice the conceptual con-tent of the transformed function, as captured by its pole diagram, and Idiscuss that topic. The relationship between the modulus of the trans-fer function and the amplitude response curve is the conceptual coreof the course. Similarly, standard treatments of generalized functions,impulse response, and convolution, typically all occur entirely withinthe context of the Laplace transform, whereas I try to present themas useful additions to the student’s set of tools by which to representnatural events.In fact, a further purpose of these notes is to try to uproot someaspects of standard textbook treatments which I feel are downrightmisleading. All textbooks give an account of beats which is mathe-matically artificial and nonsensical from an engineering perspective. Igive a derivation of the beat envelope in general, a simple and revealinguse of the complex exponential. Textbooks stress silly applications ofthe Wronskian, and I try to illustrate what its real utility is. Text-books typically make the theory of first order linear equations seemquite unrelated to the second order theory; I try to present the first or-der theory using standard linear methods. Textbooks generally give aninconsistent treatment of the lower limit of integration in the definitionof the one-sided Laplace transform, and I try at least to be consistent.A final objective of these notes is to give introductions to a few top-ics which lie just beyond the limits of this course: damping ratio andlogarithmic decrement; the L2or root mean square distance in the the-ory of Fourier series; the exponential expression of Fourier series; theGibbs phenomenon; the Wronskian; a discussion of the ZSR/ZIR de-composition; the Laplace transform approach to more general systemsin mechanical engineering; and an introduction to a treatment of “gen-eralized functions,” which, while artificially restrictive from a math-ematical perspective, is sufficient for all engineering applications andwhich can be understood directly, without recourse to distributions.These essays are not formally part of the curriculum of the course, butthey are written from the perspective developed in the course, and Ihope that when students encounter them later on, as many will, theywill think to look back to see how these topics appear from the 18.03perspective.I want to thank my colleagues at MIT, especially the engineering fac-ulty, who patiently tutored me in the rudiments of engineering: SteveHall, Neville Hogan, Jeff Lang, Kent Lundberg, David Trumper, andKaren Willcox, were always on call. Arthur Mattuck, Jean Lu, andLindsay Howie read early versions of this manuscript and offered frankadvice which I have tried to follow. I am particularly indebted toArthur Mattuck, who established the basic syllabus of this course. Healso showed me the approach to the Gibbs phenomenon included here.My thinking about teaching ODEs has also been influenced by the thepedagogical wisdom and computer design expertise of Hu Hohn, whobuilt the computer manipulatives (“Mathlets”) used in this course.They can be found at http://math.mit.edu/mathlets. Assorted er-rors and infelicities were caught by students in 18.03 and by ProfessorSridhar Chitta of MIST, Hyderabad, India, and I am grateful to themall. I am indebted to Jeremy Orloff for many recent discussions andmuch technical assistance. Finally, I am happy to record my indebted-ness to the Brit and Alex d’Arbeloff Fund for Excellence, which pro-vided the stimulus and the support over several years to rethink thecontents of this course, and to produce new
View Full Document
Unlocking...