Course Title:                              Ordinary Differential Equations

Course Code:                             MATH 2271

Level:                                         2

Number of Credits:                     3

Semester:                                   2

Prerequisite(s):                           MATH1142, MATH1151 and MATH 2273

Course Rationale

The construction of mathematical models to address real-world problems has been one of the most important aspects of each of the branches of science. It is often the case that these mathematical models are formulated in terms of equations involving functions as well as their derivatives. Such equations are called differential equations. If only one independent variable is involved, often time, the equations are called ordinary differential equations. The course will demonstrate the usefulness of ordinary differential equations for modeling physical and other phenomena.

Course Description

This is an introductory course that involves the solving of various ordinary differential equations of first and second order, as well as the solution of systems of differential equations. Methods of solution include separation of variables, various substitution techniques and use of integrating factors, undetermined coefficients, and variation of parameters. Laplace transforms, infinite series, and selected numerical methods are also incorporated. Uniqueness and existence theorems are covered.

A solid grounding in Calculus is necessary, as is knowledge of linear algebra for the theory of solution of systems of equations. For this reason, these are considered to be prerequisite courses. Prior knowledge of mathematical software (such as Maple and Matlab) will be an asset for the numerical work involved, but should not be considered to be a prerequisite.

Active learning will be achieved through assigned problem sheets allowing continuous feedback and guidance on problem solving techniques in tutorials and on myeLearning and through four major assignments. Assessment will be based on the four assignments, in-course tests and a comprehensive final examination.

Content:   (topics marked by “*” will be covered in the computer lab)

• Introduction: Basic Concepts
• Definition of an ODE, order, degree, linearity/nonlinearity, general solution, particular solution.
• Solution of an ODE that is separable/non-separable, homogeneous/non-homogeneous, exact/non-exact.
• Classification as Initial Value Problems (IVPs) and Boundary Value Problems (BVPs).
• First Order Differential Equations
• Linear equations: use of integrating factors.
• Separable equations.
• Solution of exact ODEs.
• The existence and uniqueness theorem.
• *Numerical methods; Euler's method, numerical solvers for first order ODEs.
• Second Order Linear Equations
• Definition of a second order linear differential equation.
• Solution of homogeneous linear ODEs with constant coefficients.
• Linear independence of solutions and the Wronskian.
• Methods for solving second order linear differential equations with constant and variable coefficients.
• Systems of First-Order Differential Equations
• Linear algebraic equations, linear independence, eigenvalues, eigenvectors.
• Basic theory of systems of first order linear equations.
• Homogeneous linear systems with constant coefficients.
• Complex eigenvalues.
• Fundamental matrices.
• Repeated eigenvalues.
• Nonhomogeneous linear systems.
• *Numerical solution of first order differential equations
• Power series solutions
• Review of power series
• Definition of a singular point, and ordinary point, irregular and regular singular points.
• Power series solutions near an ordinary point.
• Method of Frobenius for finding solutions about regular singular points
• Legendre and Bessel equations.
• Laplace transform method for solving ODEs
• Definition of the Laplace transform.
• The Laplace transform of special functions, derivatives and integrals.
• The inverse Laplace transform.
• Solution of ODEs via Laplace transforms.
• *Numerical Solution of ODEs of order “n”.
• *Euler's method, Improved Euler's method, Runga-Kutta methods (RK4).

Learning Outcomes

By the end of the course, students should be able to:

• Classify ODEs by order, linearity, and homogeneity, and state whether they are IVPs or BVPs.
• Solve linear, separable and exact first order ODEs.
• State the existence and uniqueness theorem.
• Utilize Euler’s method and basic numerical solvers for first order ODEs.
• Solve homogeneous second order ODEs with constant coefficients.
• Utilize the Wronksian to prove the linear independence of solutions.
• Solve second order ODEs with constant and variable coefficients.
• Express a higher order ODE as a system of first order ODEs, and solve the resulting system analytically or numerically.
• Identify the singular and ordinary points of a second order ODE, distinguishing between regular and irregular singular points.
• Use power series to solve second order ODEs about ordinary and regular singular points.
• Use Laplace transforms and their inverses to solve ODEs.
• Provide the general solution for Bessel and Legendre equations.
• Define and utilize the Laplace transform on special functions, derivatives and integrals.
• Find the inverse Laplace transform of a function.
• Utilize the Laplace transform to solve ODEs
• Use Euler’s method, the improved Euler's method, and Runga-Kutta methods (RK4) to solve ODEs of order “n”.

Teaching Methodology

This course will be delivered through a combination of informative lectures and participative tutorials. The total estimated 39 contact hours are broken down as follows: 29 hours of lectures, 7 hours of tutorials and 6 hours of computer labs (*this counts as 3 contact hours). Supporting course materials will be posted on myeLearning.

Assessment

The course assessment will be broken into two components; a coursework component worth 50% and a final exam worth 50%.

• Two course work exams (1 hour each) and each of these exams will be worth 15% of the student’s final grade.
• Four assignments (practical questions based on the theory done during lectures and computer labs) will be given throughout the semester, each worth 5%.
• The final exam will be two hours in length and will be worth 50% of the final grade.

Course Calendar

Week

Lecture Topics

Assignments

Lab/

Tutorial

1

Course Overview/Introduction

• Introduction: Basic Concepts
• Definition of an ODE, order, degree, linearity/nonlinearity, general solution, particular solution.
• Solution of an ODE that is separable/non-separable, homogeneous/non-homogeneous, exact/non-exact.
• Classification as Initial Value Problems (IVPs) and Boundary Value Problems (BVPs).

None

Tutorial #1

2

• First Order Differential Equations
• Linear equations: use of integrating factors.
• Separable equations.

Assignment 1 given

Tutorial #2

3

• First Order Differential Equations
• Solution of exact ODEs.
• The existence and uniqueness theorem.

None

Computer Lab #1

(Euler’s method, numerical solvers)

4

• Second Order Linear Equations
• Definition of a second order linear differential equation.
• Solution of homogeneous linear ODEs with constant coefficients.

Assignment 1 due

Assignment 2 given

Tutorial #3

5

• Second Order Linear Equations
• Linear independence of solutions and the Wronskian.
• Methods for solving second order linear differential equations with constant and variable coefficients.

Coursework Exam #1

No tutorial or lab this week

6

• Systems of First-Order Differential Equations
• Linear algebraic equations, linear independence, eigenvalues, eigenvectors.
• Basic theory of systems of first order linear equations.
• Homogeneous linear systems with constant coefficients.

Assignment 2 due

Assignment 3 given

Tutorial #4

7

• Systems of First-Order Differential Equations
• Complex eigenvalues.
• Fundamental matrices.
• Repeated eigenvalues.
• Nonhomogeneous linear systems.

None

Computer Lab #2

(Numerical solution of systems of first order differential equations)

8

• Power series solutions
• Review of power series
• Definition of a singular point, and ordinary point, irregular and regular singular points.
• Power series solutions near an ordinary point.

Assignment 3 due

Assignment 4 given

Tutorial #5

9

• Power series solutions
• Method of Frobenius for finding solutions about regular singular points
• Legendre and Bessel equations.

None

Tutorial #6

10

• Laplace transform method for solving ODEs
• Definition of the Laplace transform
• The Laplace transform of special functions, derivatives and integrals.

Assignment #4 due

Tutorial #7

11

• Laplace transform method for solving ODEs
• The inverse Laplace transform
• Solution of ODEs via Laplace transforms.

Coursework Exam #2

Tutorial #8

12

• Numerical Solution of ODEs
• Euler's method, Improved Euler's method, Runga-Kutta methods (RK4).

None

Computer Lab #3

(Numerical Solution of ODEs)

13

Revision

None

None

Reference Material

Essential Textbook:

• Boyce, W., E. and DiPrima, R., C. Elementary Differential Equations and Boundary Value Problems, Wiley & Sons, 2003.

Highly Recommended Textbooks

• Edwards, C., H. and Penney, D. E. Elementary Differential Equations, Pearson - Prentice Hall, 2008.
• Birkhoff, G. and Rota, G., C. Ordinary Differential Equations, John Wiley & Sons, 1989.

Online Resources: Support material will be made available via myeLearning.