# Math 319: Spring 2000

## Lecture 7: Waleffe

Day  Time  Place
TR  1:00-2:15pm B139 Van Vleck Hall

Office hours: TR: 2:15-3:15pm in VV 819.

## Textbook

Elementary Differential Equations and Boundary Value Problems, Boyce and DiPrima, 6th Edition, Required.

## Class list

You can e-mail everyone in the class by sending a message to math-319-7@lists.students.wisc.edu . This list is created and updated automatically so I cannot add or subtract you from the list. You have to be registered for this lecture and have a "students.wisc.edu" account. (a CAE account may do it too, I think).

Do you know the university policy on academic honesty? If not, check it out at the site provided by the Dean of Students. It is YOUR responsibility to follow university policies. Student dishonesty is troublesome for your classmates, your instructor, and for you. It is in everybody's best interests that you work with integrity.
(I must admit that this line was stolen from the Chem 108 page...)

Homework: (10% of total grade) Problems will be suggested weekly (posted on this page). You are encouraged to work on those in groups, helping each other out. You will have to hand in a subset of those problems written exquisitely and stapled . Presentation will be a factor. Expect several of the suggested problems (not just those you have to hand in) to find their way onto the exams. Some of the homework will involve the use of MATLAB. Prior experience with MATLAB is not required. MATLAB HELP FILES .

Exams :

## Outline

1. Introduction: definition of an ODE, basic problems (IVP and BVP), examples
2. First Order ODEs (1.5 weeks)
• a) linear equations: homogeneous and non-homogeneous
• b) nonlinear equations: e.g. , separable equations

3. More on First Order ODEs (1.5-2 weeks)
• a) direction fields
• b) existence and uniqueness theorem (for first order ODEs)
• c) the Euler scheme
• d) other numerical methods

4. Second Order Linear ODEs with Constant Coefficients (1.5-2 weeks)
• a) homogeneous equations
• b) non-homogeneous equations: method of annihilators and variation of parameters
• c) remarks on higher-order equations, linear independence and the Wronskian
• d) application to forced oscillation problems; the effect of resonances

5. The Laplace Transform (2 weeks)
• a) definition and elementary properties
• b) application to constant coefficient linear equations
• c) discontinuous forcing functions

6. First Order Systems (1.5-2 weeks)
• a) conversion of 2nd and higher order equations to systems
• b) vectors and matrices
• c) solution of linear constant coefficient systems

7. Two Dimensional Systems and the Phase Plane (2-3 weeks)
• a) classification of (equilibria for) linear systems
• b) qualitative behavior of nonlinear systems: classification of equilibria; stability
• c) applications, e.g. population models

8. Boundary Value Problems (2-3 weeks)
• a) physical origins via separation of variables from PDEs
• b) Fourier expansions
• c) eigenvalue problems
• d) more on general expansion methods

9. More on Systems (time permitting)
• a) qualitative behavior in the phase plane
• b) the dependence of equations on parameters; bifurcation
• c) chaotic solutions

10. Series Methods (time permitting)