Learning outcomes

- Recognize and solve a separable, exact, homogeneous or linear first-order ordinary differential equation using one or more appropriate methods seen in the course;

- Recognize and solve an ordinary second-order linear homogeneous or inhomogeneous differential equation using the appropriate method, including Laplace transforms.

- Use numerical and graphical methods to solve a problem with initial values.

- Recognize and use a set of mutually orthogonal functions.

- Develop a simple periodic function in Fourier series, and discuss its convergence.

- Recognize and solve a boundary condition problem involving a homogeneous partial differential equation, using separation of variables and Fourier analysis.

 

Goals

- Recognize and solve a separable, exact, homogeneous or linear first-order ordinary differential equation using one or more appropriate methods seen in the course;

- Recognize and solve an ordinary second-order linear homogeneous or inhomogeneous differential equation using the appropriate method, including Laplace transforms.

- Use numerical and graphical methods to solve a problem with initial values.

- Recognize and use a set of mutually orthogonal functions.

- Develop a simple periodic function in Fourier series, and discuss its convergence.

- Recognize and solve a boundary condition problem involving a homogeneous partial differential equation, using separation of variables and Fourier analysis.

 

Content

  1. Introduction

  2. First order differential equations

  3. Analysis and approximate methods

  4. Second order differential equations

  5. A few applications of differential equations

  6. Introduction to Fourier analysis

 

Table of contents

DIFFERENTIAL EQUATIONS

I) Reminders

II) 1st order differential equations

a.         Equations with separable variables

  1. Population models
  2. Chemical kinetics

b.         Exact differential equations

c.         Homogeneous differential equations

d.         Linear differential equations

  1.  Integrating factor method
  2. Examination of 2nd member
  3. Variation of constants
  4. Bernouilli-type equations

e.         Numerical approaches

 

III) 2nd-order differential equations

a.         With constant coefficients

  1. Homogeneous differential equations
  2. Inhomogeneous differential equations - Examination of the 2nd member
  3. Inhomogeneous differential equations - Variation of constants
  4. Inhomogeneous differential equations - Series development

b.         Coupled differential equations

c.         Eigenvalue problems

IV) Differential equations of order greater than 2

V) Laplace transform

a.         Properties

b.         Calculation

c.         Solving differential equations

VI) Some applications of differential equations in Chemistry and Geology

 

FOURIER SERIES AND TRANSFORMS

I) Introduction and reminders

II) Expression of a Fourier series

III) Fourier transform

IV) Applications


 

 

Exercices

Exercise sessions, supervised by an assistant, enable you to apply the concepts covered in the course and prepare for the exam. Exercises are available on Webcampus.

 

 

Assessment method

The exam is compulsory in January. The course is assessed by a written exam (exercises). Questions on theory may also be asked.

 

Sources, references and any support material

- Exercise syllabus (available on Webcampus)

- Bibliographical references contained in documents posted on Webcampus and/or announced during the course

 

Language of instruction

Français
Training Study programme Block Credits Mandatory
Bachelier en sciences chimiques Standard 0 3
Bachelier en sciences chimiques Standard 2 3