# Differential Equations

Class Hours: MWF 9:00-9:50

**Prerequisite: **MATH 231 (Calculus and Analytic Geometry), or equivalent

** Textbook:** Elementary
Differential Equations, 8^{th} Edition

Authors: W. E. Boyce & R. C. DiPrima, Publisher: WILEY, ISBN: 0-471-43339-X

**
About the course:**

Formulation of fundamental natural laws and technological problems in the form
of rigorous

mathematical models is given prevalently in terms of ** differential equations **
(equations that involve

functions of a single variable as well as derivatives of these functions).
Therefore, differential

equations is a most important discipline in mathematical education. Several
questions naturally arise.

o Just what is a differential equation and what does it signify ?

o Where and how do differential equations originate and of what use are they?

o Confronted with a differential equation, what does one do with it, how does
one do it, and

what are the results of such activity?

These questions indicate three major aspects of the subject: theory, method , and
application. The

purpose of this course is to introduce audience to the basic aspects of the
subject and at the same

time give a brief survey of the three aspects just mentioned. In this course, we
shall find answers to

the general questions raised above, answers which will become more and more
meaningful as we

proceed with the study of differential equations. This course provides audience
with an easy to

follow and comprehensive introduction to differential equations and is intended
to serve three

purposes:

i) to provide an accessible introduction to the world of differential equations
for students who do

not intend to specialize in this area;

ii) to provide a prerequisite course for the more specialized third and fourth
year courses in

ordinary differential equations, partial differential equations, and dynamical
systems,

iii) to provide an introduction to the discipline of Applied Mathematics,
namely, the formulation

and analysis of mathematical models of real-world phenomena. Since many models
are based on

differential equations, an introductory course in DEs provides a natural vehicle
for this purpose.

**
What you need to know:**

Success in the course depends on having a good knowledge of single variable
calculus – derivatives,

antiderivatives, qualitative curve -sketching and improper integrals, in
particular. In the final chapter,

some knowledge of linear algebra is also required – matrices, eigenvalues and
eigenvectors – but

only for the two-dimensional case.

You’ll find that the exponential function plays a major role in the subject of
differential equations,

and so it is important that you have a good grasp of exponentials and
logarithms . Somewhat

surprisingly, complex numbers are used in the course, even though all the
unknown functions are

real- valued . The reason for the appearance of complex numbers is that the roots
of real polynomials

are in general complex. So in the course you’ll find yourself using the famous
Euler formula

e^{iθ} = cosθ + i sinθ .

Some concepts from physics arise in the applications, the most important being
Newton’s Second

Law of Motion. However, in order to keep the course accessible, the background
needed for the

applications, most of which arise in everyday life, will be given in the course.

** Learning the course material:
**

Although the lectures give a discussion of the theoretical matters, the use of the computer algebra

system

**Maple**will also be used. The reason for this is that science and engineering problems in the

real world are usually of sufficient complexity that we must solve them using computer assistance.

**Class procedures:**You are expected to

o attend the lectures and take your own lecture notes,

o read the textbook and lecture notes critically,

o work out the suggested problems from the section discussed in class.

**Assignments, Tests and Exam Policy:**

o No make-ups will be allowed for assignments and tests,

o Make-up Final Exam will be given only in case of properly documented emergencies,

o Late assignments will

**not**be accepted,

o The assignments must be stapled (not folded) in the upper left corner,

o Write your name, the assignment number clearly on the top of the first page,

o You must show all your work, so it is easy to follow how you arrived at the solution.

Course Evaluation:

o Tests: (25% each) 50%

o Assignments: 10%

o Final Exam: 40%

Course Evaluation:

o Tests: (25% each) 50%

o Assignments: 10%

o Final Exam: 40%

**Preliminary course outline**

Chapters |
Topics Covered |
Suggested Problems |

1. Introduction | 1.1; 1.2; 1.3 | Will be announced |

2. First Order Differential Equations | 2.1; 2.2; 2.3; 2.4; 2.5; 2.6; 2.7; 2.8 | Will be announced |

3. Second-Order Linear Equations | 3.1; 3.2; 3.3; 3.4; 3.5; 3.6; 3.7; 3.8; 3.9 | Will be announced |

4. Higher-Order Linear Equations | 4.1; 4.2; 4.3; 4.4 | Will be announced |

5. Series Solutions of Second Order Linear Equations |
5.1; 5.2; 5.3; 5.4; 5.5; 5.6; 5.7; 5.8 |
Will be announced |

6. The Laplace Transform | 6.1; 6.2; 6.3; 6.4; 6.5; 6.6 | Will be announced |

7. Systems of First Order Linear Equations |
7.1; 7.2; 7.3; 7.4; 7.5; 7.6 |
Will be announced |

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