Introduction to Differential Equations MAT26600

• First exam: September 29 at 6:30pm in MATH 175 Worth 20 % of the final grade. Sample 1:15 first exam and now with answers (and more practice problems, but unfortunately no answers yet).
• Second exam: November 3 at 6:30pm in MATH 175 Worth 20% of the final grade. Sample 1:15 second exam (now with some answers ).
  Here are some practice problems.
• Quiz dates: September 1, September 15, October 13, November 17, December 1 Worth 10% of the final grade (in total, so about 2% each).
  
• Assignments are due weekly and is worth 15% of the final grade (collectively)
• Last lecture: December 8
  
• Final exam: TBA (Exam week is December 12-17 The final is worth 35% of the final grade.
  

• We will meet on Tuesdays and Thursdays at 12 resp. 1:30 (if you are in section 11 resp. 12). We will follow the text "Elementary Differential Equations and Boundary Value Problems" by Boyce and DiPrima (9th Ed.). Using an old edition should be perfectly o.k. EXCEPT for the homework problems - you'll have to find a friend with the current edition to make sure you have the right problems.
• The final exam covers everything from day 1. It will be a multiple choice exam. All sections of MA266 will write the exact same exam. The homework should give a good guide for what to practice. Also, you will find it quite useful to look at past exams for this course (some of which have multiple choice problems, quite similar to the exam we will write), which can be found following links from here ,

In this chapter, we will explain what a differential equation is, introduce some terminology to categorize differential equations (terms like ordinary differential equation, partial differential equation, order, linear, non-linear), talk briefly about mathematical modeling and solve a particular equation explicitly. We will also describe direction fields and use them a little to see qualitative behaviour - this can be useful if an analytic expression cannot be found (and even when they can be found).

We will discuss first order (ordinary!) differential equations (we will deal exclusively with ordinary differential equations; partial differential equations are beyond the scope of this course). Even within this relatively simple class we will have to restrict our attention to classes of equations having particular forms - linear equations, autonomous equations, exact equations (and discuss integrating factors which can sometimes be used to transform an equation into an exact equation).

We will carefully study the solution of this class of equations, mostly the case of constant coefficients. If the coefficients are constant, a very simple and complete analytic description is possible and we show how to do this (in the 'homogeneous' case). In the non-homogeneous case, we discuss some ways to get solutions from solutions to the corresponding homogeneous equation. The case of non-constant coefficients is very briefly discussed (it is discussed more carefully in Chapter 5, which we will omit in this course). Finally, two applications are discussed: simple electrical circuits, and harmonic motion (e.g. a spring satisfying Hooke's law) with damping and forcing.

We extend the results of Chapter 3 to higher order; the main point is that many of the same techniques still work with small modification.

The Laplace transform can be useful for dealing with certain forcing functions (we will concentrate on the setting of second order linear constant coefficient equations, though it can be used for higher orders and even for systems of equations) which have some discontinuity, such as a Heaviside function or even a Dirac delta function. This may arise, for example, when studying circuits where one drives the circuit with an applied voltage that is constant on intervals but jumps every once in a while (for example, a square wave). It is also an opportune time to introduce this integral transform, of which there are others (e.g. the Fourier transform) which are indispensable tools in their own right in mathematical applications beyond ordinary differential equations (PDE, number theory, signal processing, etc.).

We study coupled systems of first order linear ODE's with constant coefficients - the problem is very well understood (at least theoretically, though computationally it can be difficult) and we discuss how to solve these systems in some detail. If you took MA26500 then you will be familiar with much of what we will discuss in this chapter.
It so happens that the results of Chapters 3, 4 can be thought of as particular cases of what we will do here, because an nth order equation can be solved by writing an equivalent system of n first order equations. However, it is, perhaps, worthwhile to have done things as we did since the solutions we found earlier can be more simply described and found than the solutions of general first order systems.