Course: Séries et équations différentielles

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Collapse Expand
  Announcement
  
  Select activity Announcements
  
  Announcements Forum
  
  Due: Thursday, 23 October 2025, 12:00 AM
Select section Contact sheet

Collapse Expand
Contact sheet
Semester: 1

Course Unit: UEF-3

Course Title: Series and Differential Equations

VHS: 67.5 hours (Lecture: 3 hours, Tutorial: 1.5 hours)

Credits: 6

Coefficient: 3

Course presentation

The course ``Series and Differential Equations'' is intended for second-year undergraduate students (LMD system), Physics track, within the field of Material Sciences. It is part of the fundamental unit UEF-3 of the first semester.

This support aims to provide students with a comprehensive and structured resource in order to facilitate autonomous learning and the progressive assimilation of fundamental concepts. Through a balance of theory, detailed examples, and solved exercises, it is designed to strengthen students' analytical skills and to enable them to apply the studied methods in various contexts of physics and mathematics.

The manuscript is divided into six chapters corresponding to the

program of this fundamental unit, while strictly following the

content and the sequence of chapters as defined by the official framework

provided by the Ministry.

Course objectives

The main objective of this module is to enable students to master the techniques of differential and integral calculus essential in physics. More specifically, the targeted skills are:

1. To master the different methods of computation and evaluation of single, multiple, definite, and improper integrals.

2. To acquire analytical methods for solving differential equations.

3. To apply Laplace and Fourier transforms in solving ordinary differential equations as well as certain partial differential equations.

4. To discover numerical series, study their properties, and analyze some convergence criteria.

This course is presented by:

First name : AYADI

Last name: SOUAD

Grade: MCA

Specialty: MATHEMATICS

e-mail address: souad.ayadi@univ-dbkm.dz

Position: Module lecturer

Module evaluation: Exam*0.67 + TD*0.33

Reception of students:

Day: Sunday

Time: 10h-11h30

Location: Teachers' room

Every day on moodle
Select section Course content

Collapse Expand
Course content
Course Content:

Chapter 1: Single and Multiple Integrals (2 weeks)

Review of the Riemann integral and antiderivatives.

Double and triple integrals.

Applications to the calculation of areas, volumes, etc.

Chapter 2: Improper Integrals (2 weeks)

Integrals of functions defined on an unbounded interval.

Integrals of functions defined on a bounded interval but infinite at one endpoint.

Chapter 3: Differential Equations (2 weeks)

First- and second-order ordinary differential equations.

Elements of partial differential equations.

Chapter 4: Series (3 weeks)

Numerical series.

Sequences and series of functions.

Power series, Fourier series.

Chapter 5: Laplace Transform (3 weeks)

Definition and properties.

Application to solving differential equations.

Chapter 6: Fourier Transform (3 weeks)

Definition and properties.

Application to solving differential equations.
- Collapse Expand
  Pre-requisites
  
  -Calculus (Mathematics 1 and 2) in particular: A mastery of single-variable calculus (derivatives, antiderivatives, integrals)
  
  -Understanding vectors, dot and vector products, matrices, determinants, and eigenvalues and eigenvectors is necessary to solve certain systems of equations.
  
  -Advanced concepts: The concepts of functions of several variables, vector calculus, and the topology of R^{n} are sometimes covered in the first year and serve as a foundation for differential equations.
  
  Select activity Assess the prerequisites
  
  Assess the prerequisites Quiz
  
  Opened: Thursday, 13 November 2025, 12:00 AM
  
  Closes: Friday, 13 November 2026, 3:49 PM
  
  This test aims to assess the students' level with regard to the prerequisites and to determine whether they have mastered the key previous concepts that are necessary to follow this course.
Select section Attendance Register

Collapse Expand
Attendance Register
- Select activity Course attendance sheet
  
  Course attendance sheet Attendance Register
Select section Chapter 1

Collapse Expand
Chapter 1
- Collapse Expand
  Introduction & Chapter Objectives
  
  Introduction
  
  Integration is one of the fundamental concepts in calculus and analysis.
  It allows us to compute quantities such as areas, volumes, lengths, and other accumulated measures.
  In this chapter, we first review the basic principles of the Riemann integral and the computation of simple integrals.
  We then extend these ideas to multiple integrals, which enable the evaluation of integrals over two or three-dimensional regions.
  These tools are essential in mathematics, physics, and engineering, as they provide the foundation for solving problems involving continuous quantities such as mass, charge, and energy distribution.
  
  Chapter Objectives
  
  At the end of this chapter, students should be able to:
  
      1. Recall the definition of the Riemann integral and compute basic integrals.
  
     2.   Apply the properties of definite and indefinite integrals.
  
      3. Evaluate double and triple integrals over different domains.
  
      4. Use change of variables in multiple integrals (polar, cylindrical, and spherical coordinates).
  
      5. Apply multiple integrals to compute areas, volumes, and other physical quantities.
- Select activity Simple and Multiple integrals
  
  Simple and Multiple integrals File
- Select activity Series 1
  
  Series 1 File
- Collapse Expand
  Vedeo
  
  Select activity Integrals
  
  Integrals URL
  
  Select activity Double integral
  
  Double integral URL
  
  Select activity Triple integral
  
  Triple integral URL
- Collapse Expand
  Forum : simple and multiple integrals
  
  Select activity Simple and Multiple integrals
  
  Simple and Multiple integrals Forum
  
  Due: Saturday, 25 October 2025, 12:00 AM
- Collapse Expand
  Glossary : Simple and Multiple integrals
  
  Select activity Glossary
  
  Glossary
- Collapse Expand
  Exit Test : Simple and Multiple Integrals
  
  Select activity Exit Test
  
  Exit Test Quiz
  
  Opened: Thursday, 23 October 2025, 12:00 AM
  
  Closes: Friday, 23 October 2026, 12:00 AM
Select section Chapter 2

Collapse Expand
Chapter 2
- Collapse Expand
  Introduction & Chapter Objectives
  
  Introduction
  
  In many mathematical and physical problems, we encounter integrals that cannot be evaluated in the usual sense because the interval of integration is infinite or the integrand becomes unbounded.
  
  Such integrals are called improper integrals
  
  They extend the concept of definite integrals by using limits to assign a finite value whenever possible.
  
  In this chapter, we will study the different types of improper integrals, learn how to determine their convergence or divergence, and explore several techniques for their evaluation.
  
  Improper integrals play a crucial role in various applications, such as probability theory, Fourier analysis, and physics, where infinite processes and unbounded functions frequently appear.
  
  Chapter Objectives
  
  At the end of this chapter, students should be able to:
  
  1. Define improper integrals of the first and second kind.
  
      2. Distinguish between different types of improper integrals
  
             (infinite limits and infinite    discontinuities).
  
     3. Evaluate improper integrals using limits.
  
  4.   Determine the convergence or divergence of an improper integral.
  
  5.    Apply convergence tests such as the comparison test and the limit comparison test.
  
  6. Use improper integrals in practical applications, such as probability and physics problems.
- Select activity Improper Integrals
  
  Improper Integrals File
- Select activity Series 2
  
  Series 2 File
- Collapse Expand
  Vedeo: Improper integrals
  
  Select activity Improper integral type 1 and 2
  
  Improper integral type 1 and 2 URL
  
  Select activity Improper integral _Exercises
  
  Improper integral _Exercises URL
  
  Select activity Improper integral_1
  
  Improper integral_1 URL
- Collapse Expand
  Chat
  
  Select activity Improper integral
  
  Improper integral Forum
  
  Due: Thursday, 23 October 2025, 12:00 AM
- Collapse Expand
  Glossary
  
  Select activity Glossary : improper integral
  
  Glossary : improper integral
- Collapse Expand
  Exit Test
  
  Select activity Quiz: Improper integrals
  
  Quiz: Improper integrals
  
  Opened: Thursday, 23 October 2025, 12:10 AM
  
  Closes: Friday, 23 October 2026, 12:00 AM
Select section Chapter 3

Collapse Expand
Chapter 3
- Collapse Expand
  Introduction & Chapter Objectives
  
  Introduction
  
  Differential equations play a central role in mathematics and the applied sciences. They describe relationships between quantities and their rates of change, and therefore appear naturally in the modeling of physical, biological, and economic systems. Examples include the motion of objects, population growth, heat diffusion, wave propagation, and electrical circuits. In this chapter, we begin by studying ordinary differential equations (ODEs), focusing on first- and second-order equations and the classical methods used to solve them. We then introduce the basic concepts of partial differential equations (PDEs), which involve functions of several variables and arise in more complex phenomena such as fluid flow and heat conduction. The goal is not only to compute solutions but also to understand how differential equations are formulated, interpreted, and applied to real-world situations.
  
  Chapter Objectives
  
  At the end of this chapter, students should be able to:
  
  1. Understand the concept of a differential equation and its role in modeling physical and natural phenomena.
  
  2. Solve first-order ordinary differential equations using methods such as separation of variables, integrating factors, and exact equations.
  
  3. Solve second-order linear differential equations with constant coefficients.
  
  4. Analyze the behavior of solutions, including existence, uniqueness, and qualitative interpretation.
  
  5. Recognize basic forms of partial differential equations and understand the role of boundary and initial conditions.
  
  6. Apply differential equations to practical problems in physics, engineering, and other sciences.
- Select activity Differential Equations
  
  Differential Equations File
- Select activity Series 3
  
  Series 3 File
- Collapse Expand
  Vdeo
  
  Select activity Second order differential equation
  
  Second order differential equation URL
  
  Select activity Total Differentials
  
  Total Differentials URL
  
  Select activity Differentiability and Total Differential
  
  Differentiability and Total Differential URL
- Collapse Expand
  Forum:Differantial Equation
  
  Select activity Differential Equation
  
  Differential Equation Forum
  
  Due: Friday, 24 October 2025, 12:00 AM
- Collapse Expand
  Glossary
  
  Select activity Differential Equation
  
  Differential Equation Glossary
- Collapse Expand
  Exit Test
  
  Select activity Quiz: Differential Equation
  
  Quiz: Differential Equation
  
  Opened: Thursday, 23 October 2025, 12:00 AM
  
  Closes: Friday, 23 October 2026, 12:00 AM
- Collapse Expand
  Chat
  
  Select activity CHat
  
  CHat Forum
  
  Due: Friday, 24 October 2025, 12:00 AM
Select section Chapter 4

Collapse Expand
Chapter 4
- Collapse Expand
  Introduction & Chapter Objectives
  
  Introduction
  
  Series play an essential role in mathematical analysis and its applications, providing a powerful tool for approximating functions, solving differential equations, and modeling physical phenomena. In this chapter, we begin by studying numerical series and the methods used to determine convergence or divergence. We then extend these ideas to sequences and series of functions, where the distinction between pointwise and uniform convergence becomes important. Next, we examine power series, which allow many functions to be represented as infinite polynomials and are fundamental in calculus and mathematical physics. Finally, we introduce Fourier series, which express periodic functions as sums of sines and cosines and form the basis of harmonic analysis and signal processing.
  
  Chapter Objectives
  
  At the end of this chapter, students should be able to:
  
  1. Understand the concepts of sequences and numerical series, and determine their convergence or divergence.
  
  2. Apply common convergence tests, such as the comparison test, ratio test, root test, and integral test.
  
  3. Analyze sequences and series of functions, and distinguish between pointwise and uniform convergence.
  
  4. Work with power series, determine intervals of convergence, and represent functions as power series.
  
  5. Understand the basic principles of Fourier series and express periodic functions as trigonometric series.
  
  6. Apply series expansions to solve problems in calculus, physics, and engineering.
- Select activity Series
  
  Series File
- Select activity Series 4
  
  Series 4 File
- Collapse Expand
  Vedeo
  
  Select activity Series: All what you need to know
  
  Series: All what you need to know URL
  
  Select activity Series
  
  Series URL
  
  Select activity Series : Solved exercises
  
  Series : Solved exercises URL
- Collapse Expand
  Glossary
  
  Select activity Series : Glossary
  
  Series : Glossary
- Collapse Expand
  Forum
  
  Select activity Serie:Forum
  
  Serie:Forum
  
  Due: Friday, 24 October 2025, 12:00 AM
- Collapse Expand
  Exit Test
  
  Select activity Test your knowledge on Series
  
  Test your knowledge on Series Quiz
  
  Opened: Sunday, 16 November 2025, 5:32 PM
  
  Closes: Monday, 16 November 2026, 5:32 PM
  
  The test is attached as a PDF file
  
  Select activity Test your Knowledge on Series
  
  Test your Knowledge on Series File
Select section Chapter 5

Collapse Expand
Chapter 5
- Collapse Expand
  Introduction & Chapter Objectives
  
  Introduction
  
  The Laplace transform is a powerful technique used to convert differential equations into simpler algebraic equations. By transforming a function of time into a function of a complex variable, the Laplace transform allows us to analyze and solve problems that arise in mathematics, physics, and engineering. In this chapter, we introduce the definition of the Laplace transform, study its main properties, and learn how to compute inverse transforms. We then apply these ideas to the solution of ordinary differential equations, particularly those involving initial conditions. This method provides a systematic and efficient approach to solving problems where traditional techniques may be difficult or time-consuming.
  
  Chapter Objectives
  
  At the end of this chapter, students should be able to:
  
  1. Understand the definition of the Laplace transform and its fundamental properties.
  
  2. Compute Laplace transforms and inverse Laplace transforms for common functions.
  
  3. Use operational properties such as linearity, shifting, and convolution.
  
  4. Solve first- and second-order differential equations using the Laplace transform.
  
  5. Apply the Laplace transform to initial value problems arising in physics and engineering.
- Select activity Laplace Transform
  
  Laplace Transform File
- Select activity Series 5
  
  Series 5 File
- Collapse Expand
  Vedeo
  
  Select activity Properties of Laplace Transform
  
  Properties of Laplace Transform URL
  
  Select activity Laplace transform
  
  Laplace transform URL
  
  Select activity Laplace transform_Exercises
  
  Laplace transform_Exercises URL
- Collapse Expand
  Chat
  
  Select activity Laplace Transform
  
  Laplace Transform Forum
  
  Due: Friday, 24 October 2025, 12:00 AM
- Collapse Expand
  Glossary
  
  Select activity Glossary
  
  Glossary
- Collapse Expand
  QUIZ
  
  Select activity QUIZ
  
  QUIZ
  
  Opened: Friday, 24 October 2025, 12:00 AM
  
  Closes: Saturday, 24 October 2026, 12:00 AM
Select section Chapter 6

Collapse Expand
Chapter 6
- Collapse Expand
  Introduction & Chapter Objectives
  
  Introduction
  
  The Fourier transform is a fundamental tool in mathematical analysis used to express functions in terms of their frequency components. It plays a central role in many fields, including signal processing, physics, and engineering, due to its ability to convert complex differential equations into simpler algebraic expressions in the frequency domain. In this chapter, we introduce the definition of the Fourier transform and its inverse, explore essential properties that allow us to manipulate and simplify expressions, and learn how to compute transforms of common functions. We then apply the Fourier transform to solve differential equations and analyze systems where periodicity and frequency behavior are important
  
  Chapter Objectives
  
  At the end of this chapter, students should be able to
  
  1. Understand the definition of the Fourier transform and its inverse.
  
  2. Use fundamental properties such as linearity, scaling, shifting, and convolution.
  
  3. Compute Fourier transforms of basic functions.
  
  4. Apply the Fourier transform to solve ordinary and partial differential equations.
  
  5. Interpret the Fourier transform in the context of frequency analysis and signal processing.
- Select activity Fourier Transform
  
  Fourier Transform File
- Select activity Series 6
  
  Series 6 File
- Collapse Expand
  Vedeo
  
  Select activity Fourier Transform
  
  Fourier Transform URL
  
  Select activity Fourier transform: Introduction
  
  Fourier transform: Introduction URL
  
  Select activity Fourier Transform: Exercises
  
  Fourier Transform: Exercises URL
- Collapse Expand
  Forum
  
  Select activity Fourier Transform
  
  Fourier Transform Forum
  
  Due: Friday, 24 October 2025, 12:00 AM
- Collapse Expand
  Glossary
  
  Select activity Glossary: Fourier
  
  Glossary: Fourier
- Collapse Expand
  QUIZ
  
  Select activity QUIZ
  
  QUIZ
  
  Opened: Friday, 24 October 2025, 12:00 AM
  
  Closes: Saturday, 24 October 2026, 12:00 AM
Select section The End

Collapse Expand
The End
- Collapse Expand
  Reference
  
  Select activity Reference
  
  Reference File
- Collapse Expand
  Final Exam
  
  Select activity Final Exam
  
  Final Exam Quiz
  
  Opened: Sunday, 16 November 2025, 5:49 PM
  
  Closes: Monday, 16 November 2026, 5:49 PM
  
  The exam is atteched as a PDF file
  
  Select activity Final Exam
  
  Final Exam File
Select section New section

Collapse Expand
New section
- Collapse Expand
  New Subsection
  
  Select activity Entry -Test
- Collapse Expand
  New Subsection
- Collapse Expand
  New Subsection