announcements

• Note that Webwork Assignment 0 won't count towards the final grade
• Webwork Assignment 1 is closed on January 22
• Webwork Assignment 2 is closed on February 5
• Webwork Assignment 3 is closed on February 16
• Written assignment 1 due Monday February 24
• Reading assignment: Trench §4.1-4.2 or Zill §3.1-3.2
• Midterm solutions: Version 1, Version 2, Version 3, Version 4
• Webwork Assignment 5 is closed on March 16
• Webwork Assignment 6 is closed on March 26
• Midterm grades are posted on MyCourses
• Here is some feedback on the midterm
• Written assignment 2 due Wednesday April 2
• Webwork Assignment 7 is closed on April 11
• Written assignment 2 solutions
• class schedule

Note: This schedule is subject to revision during the term.

 Date Topic Mo 1/6 Overview of the course. Basic concepts through examples We 1/8 Falling bodies. The simplest ODE Fr 1/10 Functions. ODEs and their solutions Mo 1/13 Initial value problems. Linearity We 1/15 Using Ansatz. Separable equations [T§2.2,Z§2.2] Fr 1/17 First order linear equations: Motivating examples [T§2.1,Z§2.3] Mo 1/20 Integrating factors for linear equations [Z§2.3] We 1/22 Homogeneous and inhomogeneous linear equations [T§2.1,Z§2.3] Fr 1/24 Slope fields [T§1.3,Z§2.1] Mo 1/27 Autonomous equations [T§1.3,Z§2.1]. Euler's method [T§3.1,Z§2.6] We 1/29 Curves orthogonal to a given vector field [T§2.5,Z§2.4] Fr 1/31 Contour lines and exact equations [T§2.5,Z§2.4] Mo 2/3 Mixed partials and exact equations [T§2.5,Z§2.4] We 2/5 Examples of exactness. Bernoulli equation [T§2.4,Z§2.5] Fr 2/7 Homogeneous equations [T§2.4,Z§2.5]. Mo 2/10 Examples of ODE models [T§4.1-4.2,Z§3.1-3.2] We 2/12 Integrating factors for general equations [T§2.6,Z§2.4] Fr 2/14 Complex numbers Mo 2/17 Complex exponential and trigonometric functions. Euler's formula We 2/19 Argand plane. Exponential solutions Fr 2/21 Homogeneous 2nd order linear equations [T§5.2,Z§4.3] Mo 2/24 Linear independence. Wronskian. Fundamental set [T§5.1,Z§4.1] We 2/26 Midterm exam Fr 2/28 Abel's theorem. Variation of parameters Mo 3/10 Cauchy-Euler equation [T§5.6,Z§4.7] We 3/12 Fundamental sets. Inhomogeneous equations Fr 3/14 Undetermined coefficients [T§5.4,Z§4.4] Mo 3/17 Variation of parameters [T§5.7,Z§4.6] We 3/19 Variation of parameters (examples) Fr 3/21 Laplace transform: Linearity and scaling [T§8.1,Z§7.1] Mo 3/24 Differentiation and shift in s-space [T§8.1,Z§7.3.1,7.4.1] We 3/26 Differentiation and shift in t-space [T§8.3-8.4,Z§7.2.2,7.3.2] Fr 3/28 Solving differential equations [T§8.3-8.5,Z§7.2] Mo 3/31 Inverse Laplace transform [T§8.2,Z§7.2.1] We 4/2 Ordinary and singular points. Power series solutions [T§7.2-7.3,Z§6.2] Fr 4/4 Regular singular points. Frobenius series solutions [T§7.4-7.5,Z§6.3] Mo 4/7 Review session (optional) We 4/9 Power series solutions (examples) Fr 4/11 Frobenius series solutions (examples) Mo 4/14 Final exam (scheduled at 2pm)

course outline

[Course outline in PDF]

Exams
• The midterm exam is on Wednesday February 26, in class.
• The final exam is scheduled on Monday April 14, at 2pm.

Lectures
MWF 08:35–09:25, Stewart Biology Building S1/3

Instructor
Dr. Gantumur Tsogtgerel
Office: Burnside Hall 1123
Office hours: W 14:35–15:55, or by appointment
Email: gantumur -at- math.mcgill.ca

Recommended books
• D.G. Zill. A first course in differential equations
• D.G. Zill and W.S. Wright. Differential equations with boundary-value problems
• D.G. Zill and M.R. Cullen. Differential equations with boundary-value problems
• W. Trench. Elementary differential equations

Note: The first three books are basically the same. The first book (Zill) is a subset of the second (Zill-Wright) and the third (Zill-Cullen), and we will not cover the additional chapters that are present in Zill-Wright and Zill-Cullen. The fourth book (Trench) is a free online book, and has a similar material as in Zill.

Homework Assignments
• Webwork
• One or two written homework assignments

Webwork 15% + Written assignment 15% + max{ Midterm 10% + Final 60% , Midterm 20% + Final 50% }

Catalog description
First order ordinary differential equations including elementary numerical methods. Linear differential equations. Laplace transforms. Series solutions.

Topics to be covered
• Brief review of some relevant topics from linear algebra and calculus
• First order ordinary differential equations
• Introduction to numerical and qualitative methods
• Modelling with differential equations
• Higher order equations and systems
• Linear differential equations
• Series solutions
• Laplace transform

Prerequisite
MATH 222 (Calculus 3)

Corequisite
MATH 133 (Linear algebra and geometry)

Restriction
Not open to students who have taken or are taking MATH 325.
• midterm exam

The midterm exam will involve questions similar to the ones given in the Webwork and Written assignments, which basically means that the study topics are

• Basic concepts, separable equations
• Integrating factors for linear equations
• Slope fields, equilibrium points, Euler's method
• Exact equations
• Bernoulli's equation, homogeneous equations
• Integrating factors for general equations

I suggest below some practice problems to work on. Note that it is enough to practice from one of the books. Although these problems do not cover all of the topics listed above, they capture an important bulk of the topics. Try solving first a couple of problems from each group (There are plenty of problems, so you may not have time to solve every problem).

From Trench's book:
• §2.1: 6-11, 16-24, 30-37
• §2.4: 7-11, 22-27
• §2.5: 18-22, 33-34
• §2.6: 3-16

From Zill's book:
• §2.3: 7-24, 25-36
• §2.4: 21-26, 27-28, 31-36
• §2.5: 11-14, 21-22
• final exam

The final exam is scheduled on Monday April 14, between 2pm–5pm. Roughly speaking, the following topics will be covered.

• Integrating factors for general equations
• Second order linear equations
• Linear independence, Wronskian
• Method of undetermined coefficients
• Variation of parameters
• The Laplace transform
• Power series solutions
• Frobenius series solutions

The exam is a closed book, closed notes exam. There will be 8 problems, with difficulty of each problem comparable to that of a problem from the midterm exam. In any case, if you have done Written assignment 2, most of the problems in the exam should look very familiar.

The following formulas may come in handy if you know them off the top of your head (although they can be derived on the spot, except of course the definitions).

• Mixed partial test, formula for integrating factors
• Definition of Wronskian, Abel's formula
• Solution formula in the variation of parameters method
• Definition of and basic operational rules for the Laplace transform
• Laplace transforms of some simple functions

No reference material on the Laplace transform will be provided, since we will not go beyond simple functions such as exponentials and trigonometric functions. The absolute limit is the functions t sint and t cost.

I suggest below some practice problems to work on. Note that it suffices to practice from either one of the books. Try solving first a couple of problems from each group. The problem numbers between brackets indicate that those problems are at a basic level, so that you can skip them first and come back to them if you have trouble solving the unbracketed problems.

From Trench's book:
• §2.6: 3-16
• §5.2: (1-12, 13-17)
• §5.4: 15-19
• §5.5: 22-26
• §5.7: 1-6, 38
• §7.2: 1-8, 33-37
• §7.5: 14-25, 33
• §8.1: (2, 5)
• §8.2: (1-9)
• §8.3: 21-24
• §8.4: 7-9, 12-15, 19-22, 25
• §8.5: 2-4, 7-8

From Zill's book:
• §2.4: 31-36, 37-38
• §4.3: (1-14, 29-34)
• §4.4: 12-17, 30-33
• §4.6: 1-18
• §6.2: (3-6), 7-14
• §6.3: 15-24, 25-30
• §7.1: (24-36, 37-40, 56)
• §7.2: (1-30), 35-38
• §7.3: (1-20), 23-30, (37-48, 55-61), 66-70
• §7.4: 1-8, 11-14
• online resources

Free textbook by William Trench

Anthony Peirce's course page

Paul Dawkins' online notes

Peter Olver's page, and his introduction to Matlab and example codes

Robert Terrell's teaching page