# Math 135: General Course Outline

## Course Description

135. Ordinary Differential Equations. (4) Lecture, three hours; discussion, one hour. Requisites: courses 33A, 33B. Selected topics in differential equations. Laplace transforms, existence and uniqueness theorems, Fourier series, separation of variable solutions to partial differential equations, Sturm-Liouville theory, calculus of variations, two point boundary value problems, Green's functions. P/NP or letter grading.

General Information. Differential equations are of paramount importance in mathematics because they are equations whose solutions are functions - not numbers. Differential equations are thus widely used in mathematical models of systems where one wants to determine functional relationships. For example, the concentration of chemical reactants as a function of the time, the temperature on the surface of a heat shield as a function of position, or the size of a loan payment as a function of the interest rate. In fact, in nearly all of the courses in the physical sciences and engineering, and in many courses in the social sciences, differential equations play a fundamental role.

One of the goals of this course is to present solution techniques for differential equations that go beyond what is taught in 33B. In particular, the Laplace transform technique for solving linear differential equations is covered. This technique transforms the task of solving linear differential equations to one of solving algebraic problems. It is also a technique that can be used to solve differential equations containing generalized functions (e.g. discontinuous or Dirac delta functions). Other solution techniques include the method of Fourier series, the method of eigenfunction expansions and perturbation methods.

Another goal of this course is to introduce students to the theory of ordinary differential equations. A key part of this theory is the determination of the existence and uniqueness of solutions to differential equations. Just as it's a fact that not all algebraic equations have solutions, it's also a fact that not all differential equations have solutions. The theorems covered are especially useful, as they allow one to determine the existence and uniqueness of solutions without having to solve the differential equation.

## Textbook(s)

G. Simmons, Differential Equations with Applications and Historical Notes, 3rd Ed., McGraw-Hill.

Footnotes

1. The book does not include a review of partial fractions. Most calculus textbooks provide a suitable discussion of the technique.

2. The book only states a limited form of the Heaviside expansion theorem in problem 5 of section 53. The more general statement can be found in standard texts devoted to Laplace transforms.

3. The book provides a limited description of the use of the unit-step function and unit impulse functions. A better treatment can be found in Redheffer's book Differential Equations.

4. The proof of Theorem B is easier than Theorem A (the local existence theorem) since one doesn't have to worry about the Picard iterates leaving the domain where f(x,y) is Lipschitz. Thus, discussing and proving Theorem B before Theorem A is recommended.

5. The book glosses over some of the mathematical details required by the convergence proofs so one must supplement the material in the text as needed.

An energetic instructor may want to cover two point boundary value problems and Green's functions in more depth instead of spending the last three lectures on the calculus of variations. Alternately, one could replace the lectures on the calculus of variations with lectures on regular perturbation theory. A reference for this latter topic is Bender and Orszag, Advanced Mathematical Methods for Scientists and Engineers, Chapter 7.

Outline update: C. Anderson, 5/05

NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.

## Schedule of Lectures

Lecture Section Topics

1

General course overview.

2

17, 18

Review of solution methods and properties of solutions for linear constant coefficient equations.

3

48, 50, 51

Laplace transform. Forward transform, inverse transform. Examples of transform pairs.

4

48, 50, 51

The Laplace transform of a differential equation. The use of Laplace transforms for the solution of initial value problems.

5

48, 50, 51

Computation of the inverse Laplace transform. Partial fraction expansions revisited1.

6

49

Existence and uniqueness of Laplace transforms. Sectionally continuous functions. Exponentially bounded functions.

7

52, 53

Proof of the convolution theorem. The Heaviside expansion theorem2.

8

52, 53

The Heaviside function and Dirac distribution. Unit impulse response functions. Use of the unit impulse response function3.

9

68, 69

Existence and uniqueness theory. Examples of differential equations without unique solutions or global solutions. Lipschitz condition; determination of Lipschitz constants.

10

68, 69

Statement of a global existence and uniqueness theorem -- when f(x,y) is Lipschitz in [a,b] x [-8, 8]4. Examples of the application of the existence and uniqueness theorem.

11

68, 69

Outline of the proof of existence and uniqueness theorem. Proof preliminaries; max norm, uniform convergence, Weierstrauss M-test. Equivalence of the differential equation to an integral equation5.

12

68, 69

Picard iteration. Proof of existence and uniqueness.

13

68, 69

Local existence and uniqueness theorems. Applications of local existence and uniqueness theorems.

14

Midterm

15

33

Periodic functions and Fourier series. The inadequacy of power series approximations for periodic functions. Fourier series coefficient formulas. Examples of Fourier series.

16

35, 36

Derivation of Fourier series coefficient formulas. Fourier series for periodic functions over arbitrary intervals.

17

37

Function inner products. Orthogonal functions. Derivation of Fourier series coefficient formulas using inner products.

18

34, 38

Convergence theorems for Fourier series: Pointwise convergence.

19

34, 38

Convergence theorems for Fourier series: L2 convergence (Mean convergence).

20

40

Eigenvalues and Eigenfunctions of two point boundary value problems.

21

41

Separation of variables solution to one dimensional heat equation.

22

42

Separation of variables solution to Laplace's equation in a disk.

23

43

Sturm-Liouville problems.

24

43

Leeway

25

65, 66, 67

Calculus of Variations: Introduction.

26

65, 66, 67

Euler's differential equation for an extremal.

27

65, 66, 67

Isoperimetric problems.

28

Review