Volume: 30 lectures @ 90 minutes
Syllabus: Download Course Description (PDF)
Literature: A good overall textbook on ordinary differential equations (ODEs) is the following:
M. Braun, Differential Equations and Their Applications, Springer 1993. Free download here from within the SJTU network.
This book contains all the basic ingredients of a classic course, including some background in linear algebra, which you will no doubt recognize from Vv285. It is notable for the many fascinating applications it discusses, including art forgery, politics of nations, war and combat, biology and many other subjects where ODEs play a role.
We will also use excerpts of more diverse books for various parts of the course:
W. Walter, Ordinary Differential Equations, Springer 1998. Free download here from within the SJTU network.
This is a classic German book whose author has made a lot of contributions to the research of differential equations. You might expect that this is a tough book - and you will be correct. According to the foreword, it is intended for students in the third semester (i.e., you!) but after the first part, the majority of the book becomes quite advanced very quickly. It is useful as a reference for the initial part of our course about equations of first order. Read farther at your own peril!
K. Jänich, Linear Algebra, Springer 1994. Free download here from within the SJTU network.
At some point, we will need some more background in eigenvalue problems. We again return to our old friend (or any other of the linear algebra books recommended for Vv285) and thereby round off the linear algebra course that we have embedded in the Honors Mathematics sequence.
M. Taylor, Partial Differential Equations I (Basic Theory), Springer 2011. Free download here from within the SJTU network.
Although this book is concerned primarily with the theory of PDEs, it contains a very succinct (and hard!) review of
ordinary differential equations. We will refer to it for some proofs in the theory of systems of ODEs that we skip in our course. I strongly recommend that you read the first, introductory section, "1. The derivative" and especially the exercises to that section. It will give you an entirely new perspective on some of the material we learned in Vv285.
E. Stein and R. Shakarchi, Complex Analysis, Princeton University Press 1996. Link to Publisher's Description.
Available for purchase here.
Smack in the middle of our course we will stumble into the fascinating and wondrous world of functions of complex variables. Why? You will have to wait and see... This book is highly recommended because it heads nearly directly to the topic we need ourselves, the theory of residues. It is also very well-written, contains numerous examples and there is a lot left to explore after we have finished the first part. (It should also not go unmentioned that Elias Stein is a very well-known and respected researcher in mathematics.)
Prerequisites: Vv285
Background and Goals: The sequence Honors Math Vv186-285-286 is an introduction to calculus at the honors level. It differs from the Applied Calculus sequence in that new concepts are often introduced in an abstract context, so that they can be applied in more general settings later. Most theorems are proven and new ideas are shown to evolve from previously established theory. Initially, there are fewer applications, as the emphasis is on first establishing a solid mathematical background before proceeding to the analysis of complex models.
The present course focuses on ordinary differential equations and their applications.
Key Words: Ordinary differential equations (ODEs) of first order; systems of first-order equations; the existence and uniqueness theorem of Picard-Lindeloef; eigenvalue problems, diagonalization and the spectral theorem; Jordan normal form; application to linear systems of first-order equations; linear second-order equations; elements of complex analysis and residue theory; the Laplace transform and its inverse with applications to ODEs; power series solutions of ODEs by the Frobenius method; Bessel’s and Legendre’s differential equations; the Weierstrass approximation theorem and generalized Fourier series; introduction to the classical partial differential equations of physics and some basic solutions by separation of variables.
Detailed Content: This course consists of four distinct parts. In the first part, we will discuss some basic integrable single first order ordinary differential equations. In particular, we will look at several types of explicit and implicit equations, including homogeneous, separable, linear, Bernoulli, Ricatti, Clairaut and d'Alembert equations. We will also look at some concrete modeling examples, such as C-14 dating (using the differential equation for unrestricted growth or decay to zero) and population models (using various flavors of the logistic equation).
In the second part, we will discuss systems of first order equations. After proving a general existence and uniqueness theorem (which also has practical applications) we will introduce some background in linear algebra, in particular eigenvalue problems and matrix similarity. This part rounds off the linear algebra that was treated in the Vv285 course and thus completes our "embedded" linear algebra course. Using these techniques, we will be able to solve constant-coefficient linear systems exactly. Next, we will give a brief introduction to general systems of equations, which touches upon the theory of dynamical systems.
The third part is devoted to integration techniques for solving second-order differential equations, with an emphasis on the Laplace transform. In order to get a full grasp of the inverse transform, it is necessary to learn about residue calculus in elementary complex analysis. Since this is also useful elsewhere, and the general concepts of complex analysis will pop up again in more advanced courses, we will devote several lectures to an introduction to complex analysis. Following this, we are able to introduce the Heaviside operator calculus for solving differential equations and from that deduce the Laplace transform technique. Since the Dirac "delta function" is used frequently in applications, we will, moreover, give a brief introduction to locally convex spaces and the space of tempered distributions as dual to the Schwartz space of functions of rapid decrease.
In the last part of the course we discuss series-based solutions. the power-series-based Frobenius method leads us to the Bessel functions, which turn out to have a wide range of applications in physics and engineering. We discuss the problem of a hanging chain, self-buckling of a column, diffraction by a circular aperture and more. Series solutions based on trigonometric functions lead to Fourier series, which we view in the general context of orthogonal functions. We also apply this theory to orthogonal Bessel functions and Legendre polynomials and use these to treat some classical partial differential equations by separation of variables.
Alternatives: Vv256 (Applied Calculus IV) is an applications-oriented course, which covers much of the same material.
Overview:
The following textbooks are referenced below:
[B] Braun, M., Differential Equations and their Applications
[W] Walter, W., Ordinary Differential Equations
[J] Jänich, K., Linear Algebra
[S] Stein, E. M. and Shakarchi, R., Complex Analysis
[K] Korenev, B., Bessel Functions and their Applications
Lecture | Lecture Subject | Textbook |
---|---|---|
1 | Introduction and Explicit First-Order ODEs | [W] Ch. 1, § 1 |
2 | Separable Equations | [W] Ch. 1, § 2 |
3 | Linear and Transformable Equations | [W] Ch. 1, § 2 |
4 | Integral Curves and Implicit Equations | [W] Ch. 1, §§ 3,4 |
5 | Systems of First-Order ODEs | [W] Ch. 3, § 10 |
6 | The Eigenvalue Problem | [J] Ch. 9 |
7 | The Spectral Theorem for Self-Adjoint Matrices | [J] Ch. 10 |
8 | The Jordan Normal Form | [J] Ch. 11.3 |
9 | Linear Systems of First-Order ODEs | [B] Sec. 3.11, 3.12 |
10 | Vibrations | [B] Sec. 2.6 |
11 | First Midterm Exam | |
12 | Complex Analysis | [S] Ch. 1 |
13 | Properties of Holomorphic Functions | [S] Ch. 2 |
14 | Singularities and Poles | [S] Ch. 3 |
15 | Residue Calculus | [S] Ch. 3 |
16 | The Heaviside Operator Method | -- |
17 | The Laplace Transform | [B] Sec. 2.9-2.13 |
18 | The Laplace Transform | [B] Sec. 2.9-2.13 |
19 | The Fourier Transform | [S] Ch. 4 |
20 | Second Midterm Exam | |
21 | Power Series Solutions to Second Order ODEs | [B] Sec. 2.8 |
22 | Power Series Solutions to Second Order ODEs | [B] Sec. 2.8 |
23 | Applications of Bessel Functions | [K] |
24 | Applications of Bessel Functions | [K] |
25 | Orthonormal Functions | [B] Ch. 5 |
26 | Fourier Series | [B] Ch. 5 |
27 | Boundary Value Problems | [B] Ch. 5 |
28 | The Wave and Heat Equations | [B] Ch. 5 |
29 | The Wave and Heat Equations | [B] Ch. 5 |
30 | Final Exam |
Lecture Slides (2018 version; PDF):
(There will always be minor modifications from one iteration of the course to the next; if you are presently taking the course, it is not advisable to print out all these slides at the beginning of the term.)
Assignments (2015 version; PDF files):
(There will always be minor modifications from one iteration of the course to the next; if you are presently taking the course, your assignments may differ from these.)