# Introduction to Differential Equations

Effective Date:
Course
Discontinued
No
Course Code
MATH 2421
Descriptive
Introduction to Differential Equations
Department
Mathematics
Faculty
Science & Technology
Credits
3.00
Start Date
End Term
Not Specified
PLAR
No
Semester Length
15
Max Class Size
35
Contact Hours
4 hours lecture + 1 hour tutorial
Method Of Instruction
Lecture
Tutorial
Methods Of Instruction

Lectures, problems sessions, assignments (written and/or Maple).

Course Description
This course is an introduction to ordinary differential equations. Topics include the solution of first- and higher order differential equations, power series solutions, Laplace transforms, linear and non-linear systems, stability and applications.
Course Content
1. First-Order Differential Equations: linear, separable, autonomous and exact, existence and uniqueness of solutions, numerical methods and applications.
2. Higher Order Differential Equations: reduction of order, homogeneous linear equations with constant coefficients, nonhomogenous equations and undetermined coefficients, variation of parameters
3. Equations with Variable Coefficients: Cauchy-Euler equations and power series solutions about ordinary and singular points, Bessel and Legendre Equations
4. Laplace Transforms and applications
5. Systems of Linear Differential Equations: systems of homogeneous and nonhomogeneous first-order equations, reduction of higher-order linear equations to normal form
6. Non-linear Systems and Stability:  solutions and trajectories of autonomous systems, stability of critical points
Learning Outcomes

Upon completion of MATH 2421 the student should be able to:

• identify an ordinary differential equation and classify it by order or linearity
• determine whether or not a unique solution to a first-order initial-value problem exists
• understand differences between solutions of linear and non-linear first-order differential equations
• recognize and solve linear, separable and exact first-order differential equations
• use substitutions to solve various first-order differential equations (optional)
• recognize and solve autonomous first-order differential equations, analyze trajectories, and comment on the stability of critical points
• use the Euler method to approximate solutions to first-order differential equations
• model and solve application problems using linear and non-linear first-order differential equations, including, but not limited to, topics such as: growth and decay, series circuits, Newton’s Law of Cooling, mixtures, logistic growth, chemical reactions, particle dynamics
• determine whether or not a unique solution to a linear nth-order initial-value problem exists
• determine whether or not a set of solutions to a differential equation are linearly dependent or independent using the Wronskian
• use reduction of order to find a second solution from a known solution
• solve homogeneous linear equations with constant coefficients
• express linear differential equations in terms of differential operators (optional)
• use the method of undetermined coefficients to solve nonhomogeneous linear differential equations for which the nonhomogeneous term can be annihilated
• solve nonhomogeneous linear differential equations using variation of parameters
• solve nonhomogeneous linear differential equations using Green’s functions (optional)
• model, solve and analyze problems involving mechanical and electrical vibrations using second-order linear differential equations
• determine ordinary and singular points of linear differential equations
• recognise and solve Cauchy-Euler equations
• use power series techniques to solve linear differential equations in the neighbourhood of ordinary points
• use the method of Frobenius to solve linear differential equations about regular singular points
• use series methods to solve Bessel, modified Bessel and Legendre equations (optional)
• state the definition of the Laplace transform of a function and the sufficient conditions for its existence
• determine the Laplace transforms for basic functions, derivatives, integrals and periodic functions and find inverse transforms
• use the convolution theorem and translation theorems to find Laplace transforms and their inverses
• use Laplace transforms to solve initial value problems, integral equations and integro-differential equations
• solve systems of differential equations using differential operators or Laplace transforms (optional)
• reduce higher-order linear differential equations to first-order systems in normal form
• solve systems of homogeneous first-order linear differential equations using matrix methods
• solve systems of nonhomogeneous linear first-order differential equations
• model and solve application problems using systems of first-order linear differential equations, including, but not limited to, topics such as: parallel circuits, mixtures, chemical reactions, particle dynamics, competition models
• find trajectories associated with, determine critical points of, and perform phase plane analyses for simple autonomous linear and non-linear systems of equations
Means of Assessment

Evaluation will be carried out in accordance with Douglas College policy.  The instructor will present a written course outline with specific evaluation criteria at the beginning of the semester.  Evaluation will be based on some of the following criteria:

Tutorials: 0-10%
Tests: 20-70%
Assignments/Group work: 0-20%
Attendance: 0-5%
Final exam: 30-40%

Textbook Materials

Consult the Douglas College bookstore for the current textbook. Examples of textbooks under consideration include:

A First Course in Differential Equations with Modeling Applications, Zill, Dennis G., Brooks-Cole, current edition

Elementary Differential Equations, Johnson and Kohler, Pearson, current edition

Elementary Differential Equations and Boundary Value Problems, Boyce and DiPrima, Wiley, current edition

Prerequisites

MATH 1220 and MATH 2232 or instructor permission