Math 255: Differential Equations
2024-2025, Fall
Course Syllabus
Instructor Prof. Dr. Nassar Aghazadeh Departments Comp. Eng., Env. Eng,
E-mail nasseraghazadeh@iyte.edu.tr Food Eng.
Address Office 124, Dept. of Mathematics Venue Prof. Dr. Erdal Saygın
Lecture Hall
Day & Time 13.30–15.15, Tuesday
13.30–15.15, Thursday
Instructor Assoc. Prof. Dr. Gökhan Şahan Departments Bioeng., Elec.-Elec. Eng.,
E-mail gsahan@iyte.edu.tr Mat.-Sci. Eng., Physics
Address Office 211, Dept. of Mathematics Venue Prof. Dr. Erdal Saygın
Lecture Hall
Day & Time 10.45–12.30, Tuesday
10.45–12.30, Thursday
Instructor Dr. Kemal Cem Yılmaz Departments Chem. Eng., Civil Eng., Energy
E-mail cemyilmaz@iyte.edu.tr Sys. Eng., Mech. Eng., Phot.
Address Office 116, Dept. of Mathematics Venue Prof. Dr. Erdal Saygın
Lecture Hall
Day & Time 15.30–17.15, Tuesday
15.30–17.15, Thursday
Course Description
A first step to construct a mathematical model that describes a physical phenomenon is to observe and
understand the movement process. A movement action, brings along the notion of rate of change. Therefore, a
way to establish a mathematical model for a movement process is to write a mathematical relation that relates
the state and its rate of changes. If the movement is happening continuously, then a better way to express the
mathematical relation that is more consistent with the reality is to take into account the instantaneous rate of
changes, so called derivatives. Then, the resulting mathematical model becomes a relation involving the state
and its derivatives, called differential equation.
In this one semester introductory course, the goal is to give fundamental concepts of differential equations to
the students. Content of the course mainly covers classification of differential equations, strategies for deriving
solutions to various class of differential equations or systems of differential equations, analyzing them from
geometrical point of view and establishing mathematical models of some basic real world applications.
Learning Outcomes
Upon successfully completing this course, it is expected that students have following outcomes:
Be able to identify and classify a differential equation with respect to its order and linearity. Be able to
determine whether it is homogeneous or nonhomogeneous.
Understand that many problems from real world can be modeled by differential equations.
Be able to develop the slope field for a first-order differential equation to illustrate geometrical view of
the general solution.
Understand importance of existence and uniqueness of a solution to a given initial-value problem.
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Math 255 Differential Equations, Syllabus
Understand that different class of differential equations have different type of solving strategies. Be able to
determine and apply appropriate solving strategies to derive an analytical solution for a given differential
equation.
Be able to express a higher-order linear differential equation as a system of first-order differential equa-
tions.
Be able to solve system of first-order differential equations. Be able to analyze asymptotic behaviour of
solutions by investigating the associated coefficient matrix of the system of differential equations.
Be able to use Laplace transform to solve second-order linear differential equations with discontinuous
source functions.
Prerequisites
You should be familiar with the following subjects of first-year calculus class.
Limits of single variable functions
Derivatives of single variable functions and differentiation techniques
Integrals of single variable functions and integration techniques
Notion of power series
If you feel inadequate on at least one of these subjects, it is recommended for you to consult following sections
of Thomas G, M. Weir, J. Hass, Thomas’ University Calculus. Early Transcendentals, Pearson Education, in
SI units 15th global edition:
Trigonometric and exponential functions, inverse functions (Section 1.3–1.5).
Limits and continuity (Sections 2.2, 2.4, 2.6).
Limits involving infinity, asymptotes (Sections 2.5, 4.5).
Instantaneous rate of changes: Derivatives (Sections 2.1, 3.1).
Differentiation rules and implicit differentiation (Sections 3.3, 3.5–3.9).
Linearization (Section 3.11).
Antiderivatives, definite integrals and fundamental theorem of calculus (Sections 4.8, 5.3, 5.4).
Integration techniques (Sections 5.5, 8.1–8.5).
Improper integrals (Section 8.8)
Power series, Taylor and Maclaurin series (Sections 9.7, 9.8).
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Math 255 Differential Equations, Syllabus
Course Materials and Resources
Communication and main resources. All announcements will be posted via Microsoft Teams. Please use
the code
6gkr1db
to enroll our Team Room “M255 Differential Equations 2024 Fall ” in Microsoft Teams. In this platform you
can find class notes, video lectures involving problem solvings, graphics and animations that will improve your
learning and various type of online materials.
Textbooks. There are huge variety of textbooks written on differential equations. You can use following
textbooks as a resource:
1. W.E. Boyce, and R.C. Diprima, (2010). Elementary differential equations and boundary value problems.
10th edition. John Wiley & Sons Inc., New York.
2. R.K. Nagle, E.B. Staff and A.D. Snider, (2018). Fundamentals of differential equations. 9th edition.
Pearson, London.
3. S.L. Ross (1984). Differential equations. 3rd edition. John Wiley & Sons, Inc., New York.
Electronic Sources. You can use the following set of video lectures that will aid you to improve your learning
on differential equations.
4. Differential Equations - Video Lectures
MATLAB ® Throughout the semester, we will share simulations and graphics produced by MATLAB, to
support your learning on differential equations.
5. MATLAB
We will provide several numerical algorithms that gives approximate solutions to differential equations as well
as some basic video tutorials. These tutorials will not only improve your learning but also give an elementary
idea on how a differential equation can be solved numerically and therefore its solution can be simulated. As
an IZTECH student, you have a campus wide license for MATLAB. Click
https://bidb.iyte.edu.tr/matlab/
to see the instructions for installing the product on your device.
Grading Policy and Expectations
Exams. There will be 3 Midterm Examinations 20% of your total grade each and 1 Final Examination %40
of your total grade.
WebWork. Each week we will share a set of homework via WebWork. Solving these exercises and returning
your works are not mandatory but it is strongly expected. In this way, we intend you to make a weekly study
schedule to study regularly (rather than from exam to exam). A regular schedule will help to strengthen your
memory, improve your focus, manage your time which all of these play effective roles in your path to success.
To encourage you to do your homeworks, we assure you that some of those exercises will exactly appear in the
exams.
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Math 255 Differential Equations, Syllabus
Video Lectures. Each week we will share some tutorial videos. These videos involve problem-solvings related
to the subjects covered during lectures of that week. This activity is intended not only to reinforce related
subjects and get you prepared for your next lecture but also to help you to improve your self-learning skills.
Attendance. Attendance to lectures is not mandatory but it is strongly expected. Learning is not just about
passively watching offline recordings or reading texts. If you want to make your learning process efficient, you
should be present in the classes, not just physically but also participate interactively. In this way, you can
benefit from the teacher’s guidance as well as from your classmates through interacting with them.
Based on the above criteria, your Total Grade will be evaluated by the following formula:
Total Grade = 20% of M-1 + 20% of M-2 + %20 of M-3 + %40 of F.
Your Letter Grade will be evaluated according to your Total Grade. Unless indicated otherwise, evaluation of
the letter grades will based on the catalog system declared in IZTECH Graduate Education Regulations.
Total Grade Letter Grade
90–100 AA
85–89 BA
80–84 BB
75–79 CB
70–74 CC
65–69 DC
60–64 DD
50–59 FD
0–49 FF
Office Hours
Tutoring Center (at Dept. of Mathematics). Tutoring center is the place where you are able to ask
your questions during the certain hours of each week to the assigned teaching assistant. Exact times will be
announced via Microsoft Teams.
Important Dates
Exams will be held in the specified weeks below.
Midterm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4th week
Midterm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8th week
Midterm 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12th week
Final Exam . . . . . . . . . . . . . . . . . . . . . . During the final week
Final Week . . . . . . . . . . . . . . . . . . . 6th of Jan. – 17th of Jan.
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Math 255 Differential Equations, Syllabus
Exact dates and detailed information will be announced via Microsoft Teams. See also Academic Calendar.
Course Outline
Chapter 1: Introduction
§1.1 Introduction to Differential Equations
§1.2 Classification of ODEs and Their Solutions
Chapter 2: First Order Ordinary Differential Equations
§2.1 Separable Equations
§2.2 Linear Equations
§2.3 Nonlinear Equations: Bernoulli and Ricatti Differential Equations
§2.4 Modeling with First-Order Equations
§2.5 Exact Equations and Integration Factors
§2.6 Existence and Uniqueness Theorem
Chapter 3: Second-Order Ordinary Differential Equations
§3.1 Introduction
§3.2 Linear Homogeneous Equations: Existence and Uniqueness
§3.3 Constant Coefficient, Homogeneous Equations
§3.4 Free Mechanical Vibrations: The Mass–Spring Oscillator
§3.5 Constant Coefficient, Nonhomogeneous Equations
§3.6 Forced Mechanical Vibrations
§3.7 Variable Coefficient Equations
Chapter 4: Higher-Order Ordinary Differential Equations
§4.1 General Theory
§4.2 Homogeneous Differential Equations with Constant Coefficients
§4.3 Operator Methods for Finding Particular Solutions
§4.4 The Method of Undetermined Coefficients
§4.5 The Method of Variation of Parameters
Chapter 5: System of Ordinary Differential Equations
§5.1 Introduction
§5.2 Review on Matrices and System of Linear Equations
§5.3 Fundamentals on First Order Linear Equations
§5.4 Complex Eigenvalues
§5.5 Repeated Eigenvalues
§5.6 Fundamental Matrices
§5.7 Nonhomogeneous Linear Systems
Chapter 6: Laplace Transformation
§6.1 Definition and Properties of Laplace Transformation
§6.2 Laplace Transform Solution of Constant Coefficient Initial Value Problem
§6.3 Discontinuous Forcing Functions
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