#### Introduction

This is a first course in ordinary differential equations and classical solution methods. The emphasis will be on classes of ordinary differential equations for which explicit (or implicit) closed form solutions can be expressed, and finding those solutions. Thus we will mainly be covering linear ordinary differential equations.

The course will have a quite applied flavour, with the emphasis on techniques for solving differential equations rather than stating and proving theorems. It is an honours course though, so expect theory to come with the methods.

The material in this course complements the material in math376. Most nonlinear differential equations (and in particular those with chaotic solutions) do not have closed form solutions. Math376 studies techniques for investigating the behaviour of the solutions of differential equations in the cases where neat formulae for the solutions cannot be written down. Both books have a chapter that gives an introduction to the material in math376, so if you are planning on taking both math325 and math376 I would suggest taking this course first or both first in the same semester, if your schedule allows. However, formally math325 is not a pre- or co-requisite for math376.

Math-325 is a prerequisite though for Math-375 (partial differential equations), Math-437 and Math-574. Math-375 is in turn a prerequisite for Math-579 (which explores numerical solution of differential equations) as well as Math-580.

#### Topics

The contents and order of the course will be roughly:

• Introduction: Basic, terminology, classification.
• First Order Equations: Integrating Factors, separable equations, linear and nonlinear equations, exact equations, existence and uniqueness.
• Second and Higher Order Linear equations: Constant Coefficient homogeneous equations, roots of the characteristic equation, Wronskians, fundamental and general solutions. Cauchy-Euler Equations. Nonhomogeneous equations, undetermined coefficients, variation of constants (parameters), reduction of order.
• Series Solutions: Regular Points, Regular Singular Points and Frobenius method.
• The Laplace Transform: Definitions and properties, solving initial value problems, Discontinuities and Impulses, Convolutions.
• Linear Systems of ODEs: Solutions and stability.
• Introduction to numerical methods and nonlinear differential equations.

We may cover some things not explicitly on the list, in particular I will highlight applications through examples of differential equations throughout the course, and we may cover some parts in more detail than others, depending on time.

#### Audience/Prerequisites

The course is intended for honours students in Mathematics, Physics and Engineering programs. The only prerequisite is Math 222 or equivalent. However, it is an Honours course that will be taught at an honours level and will assume a corresponding level of mathematical sophistication. Only take this course if you are registered or considering registering in an honours program or have the standing to do so. Students whose program requires them to take math-315 should take that course. Since this is a first course in ordinary differential equations, it is not open to students who have already taken such a course, thus it is not open to students who have taken math-263 or math-315.

#### References

There is not a required textbook. In previous years/semesters the following books have been used:

• Elementary Differential Equations, 10th edition, by W.E. Boyce and R.C. DiPrima (but previous editions are just as good). There are two versions of this book. The "with Boundary Value Problems" version has an extra chapter and is more expensive.
• Elementary Differential Equations, 6th edition, by C.H. Edwards and D.E. Penny, again, previous editions are just as good, and again this book has two versions.
• Differential Equations and Boundary Value Problems, 8th edition, by D.G Zill and W.S Wright.