# MATH 334 - Introduction to Differential Equations (Fall 2014)

## Time and Location

Time: MWF 12:00 - 12:50 pm
Room: SAB 325

## Instructor

Dr. Arno Berger (CAB 683, berger@ualberta.ca)

## Office hours

MWF 2:30 - 4:00 pm

## Textbook

Stanley J. Farlow, An Introduction to Differential Equations and their Applications, Dover 2006.

You can obtain a copy of this inexpensive textbook from the University bookstore, the Dover website and many online bookstores. Despite being a great text, Farlow's book does contain a number of typos and misprints; if you spot one that is not listed here, you may want to report it (to your instructor, the errata website, or both).

## Syllabus

Please see this PDF document for all relevant details concerning MATH 334. (An abbreviated version of this document will be handed out in class.)

## Material covered in class (Course Diary)

I plan to keep an up-to-date list of the topics, examples etc. covered in class. Unless stated otherwise, reference numbers refer to our textbook, S. Farlow An Introduction to Differential Equations and their Applications, henceforth referred to as [F].

Lecture # Date Material covered / special events Remarks/ additional material
WELCOME TO
MATH 334 !!

1 Wed
3
Sep
Introduction - What is a differential equation (DE)? What is a solution of a DE? First examples. [F, Ch.1]
2 Fri
5 Sep
Examples: growth/decay of a single species; free fall; crossing a river by boat; vibrations of a long elastic string.
Ordering principles: ODE vs. PDE; order; linear vs. nonlinear; explicit vs. implicit.
[F, Ch.1]
3 Mon
8 Sep
Making solutions unique: Initial value problems (IVP) and Boundary value problems (BVP). First-order ODE: some aspects of Picard's Theorem. [F, Ch.1 and 2]
4 Wed
10 Sep
Examples of separable equations. Logistic growth. [F, Sec.2.2]
Your first assignment of MATH 334 homework is waiting for you, see the box on the left for details.
5 Fri
12 Sep
Linear first-order equations.
Examples.
[F, Sec.2.1]
6 Mon
15 Sep
Exact equations. Examples. [F, p.83]
7 Wed
17 Sep
Homogeneous equations. Examples. [F, p.43]
8 Fri
19 Sep
Equations of the form y'=f(ax+by +c).
First-order equations in disguise.
[F, p.43]
9 Mon
22 Sep
Linear second-order equations.
Basic concepts: principle of superposition, linear (in)dependence, Wronski determinant.
[F, Sec.3.1-2]
10 Wed
24 Sep
Why are linearly independent solutions important?
Reduction of order.
[F, Sec 3.3]
11 Fri
26 Sep
How to solve an inhomogeneous equation.
Homogeneous equations with constant coefficients.
[F, Sec.3.4-6]
12 Mon
29 Sep
Inhomogeneous equations:
The Undetermined Coefficients method.
[F, Sec.3.7]
13 Wed
1 Oct
Inhomogeneous equations:
The Variation of Parameters method.
[F, Sec.3.8]
14 Fri
3 Oct
An important application:
Free and forced vibrations.
[F, Sec.3.10]
15 Mon
6 Oct
Forced vibrations. Resonance. [F, Sec.3.11]
16 Wed
8 Oct
Linear higher-order equations. [F, Sec.3.12.]
Fri
10 Oct
Do not forget:
Midterm test #1.
Mon
13 Oct
No class. Happy Thanksgiving !
17 Wed
15 Oct
Examples. Euler equations. [F, Sec.3.12]
18 Fri
17 Oct
A quick reminder of power series. [F, Sec.4.1]
19 Mon
20 Oct
Solving linear ODE near ordinary points. [F, Sec. 4.2]
20 Wed
22 Oct
Illustrating the key idea of power series solutions.
Ordinary vs. singular points.
[F, Sec.4.2]
21 Fri
24 Oct
Theoretical background. An analytic version of Picard's Theorem. [F, Sec.4.3]
22 Mon
27 Oct
Power series solutions near an ordinary point. [F, Sec.4.3]
23 Wed
29 Oct
Modified power series solutions near a regular singular point. [F, Sec.4.4]
24 Fri
31 Oct
Method of Frobenius.
Example.
[F, Sec.4.4]
25 Mon
3 Nov
Special equations and functions: Airy, Hermite, Chebyshev, ... [F, Ex.4.2.3]
26 Wed
5 Nov
... and Bessel.

Laplace Transform. Basic properties.
[F, Sec. 4.5, 5.1, 5.2]

An excellent online resource for special funcions.
27 Fri
7 Nov
Further properties of Laplace Transform.
Review of Partial Fraction Expansion.
[F, Sec. 5.2, 5.3]
Mon
10 Nov
No class.
(Remembrance Day Break)
28 Wed
12 Nov
Using Laplace Transform to solve IVP. [F, Sec. 5.4]
Fri
14 Nov
Do not forget:
Midterm test #2.
29 Mon
17 Nov
Discontinuous and delayed signals. [F, Sec. 5.5]
30 Wed
19 Nov
Periodic signals. [F, Sec. 5.6]
31 Fri
21 Nov
Examples. Convolution and its properties. [F, Sec. 5.8]
32 Mon
24 Nov
Implusive forces. The Dirac delta function. [F, Sec. 5.7]
33 Wed
26 Nov
An introduction to systems of ODE. [F, Sec. 6.1]
34 Fri
28 Nov
Solving linear systems by means of eigenvalues and -vectors.
[F, Sec. 6.4, 6.5]
35 Mon
1 Dec
Solving linear systems by means of Laplace transform. [F, Sec. 6.7]
36 Wed
3 Dec
Examples.
Final examination details.
[F, Sec. 6.7]
Good bye and good luck !!
Fri
12 Dec
Special MATH 334 office hours:
10am - 3pm in CAB 683.
Mon
15 Dec
Special MATH 334 office hours:
10am - 3pm in CAB 683.
Tue
16 Dec
Final exam !

Please see box on the left for details.
Good luck !!

#### Homework

Weekly homework assignments will be posted here. Unless stated otherwise, the deadline for homework submission is 2:00 pm on Wednesday. Please submit your solutions into the designated MATH 334 assignment box on the third floor of CAB.

#### Midterm tests

There will be two midterm tests during the semester. These tests will be held in class, on Friday, October 10, 2014, and on Friday, November 14, 2014.

Some details about the first midterm:

• Duration: 50 minutes.
• Material covered: up to, and including the variation of parameters method for linear second-order ODE
• NO textbooks, notes, calculators, formula sheets etc.!
• NO cell-phones, i-pods, or other electronics!
• Please bring a valid ID with you.
• Good luck!

To help you prepare for the midterm test, here is a practice version. The real test will be very similar.

And now for the real thing ...

Some details about the second midterm:

• Material covered: everything up to, and including series solutions for linear ODE (Chapter IV in class), with a clear emphasis on LINEAR second-order equations (Chapters III and IV).
• Procedures are the same as for the first midterm (see above).
• Good luck!

To help you prepare for the second midterm test, here is another practice version. The real test will be very similar.

Again, here is what really happened ...

#### Final exam

The final exam will be held on Tuesday, December 16, 2014 at 2:00 pm in CCIS L 1-140.

Some details concerning the final:

• Duration: 120 minutes.
• Material covered: The entire semester of Math 334, but with a focus on material discussed in class after the second midterm (i.e., Laplace transform and linear systems).
• NO calculators, formula sheets etc.!
• NO cell-phones, i-pods, or other electronics!
• Please bring a valid ID with you.
• Good luck!

To help you prepare for the final exam, here is a practice version. You can expect the real exam to be very similar.

#### Other material

Your integration skills are a bit rusty? The Math and Applied Sciences Centre is running a Review of Integration Techniques.