## Mathematics 235

### Introduction to Differential Equations for Engineers

Credits:4

Length of Course: 14 weeks

Classrom Hours per Week:5

Prerequisite: Mathematics 213

Co-requisites: Mathematics 252

Text: Elementary Differential Equations and Boundary Value Problems. Boyce & DiPrima. 10th Edition.

### Course Description:

This course is an introduction to differential equations for students who intend to study engineering. Besides first and second order ODEs, linear systems and Laplace transforms, the syllabus also includes Fourier series and some basic partial differential equations.

### Course Outline:

Length Description

Week 1

Linear, first order ODEs:

• Introduction
• Homogeneous, linear, constant coefficient, first-order ODEs
• Inhomogeneous, linear, constant coefficient, first-order ODEs
• Integrating factors for non-constant coefficient, linear, first-order ODEs

Week 2

Nonlinear, first-order ODEs:

• Separable first-order ODEs
• Bernoulli ODEs
• Homogeneous ODEs

Week 3

Nonlinear, first-order ODEs:

• Existence and uniqueness (linear vs nonlinear ODEs
• Autonomous first-order ODEs and stability

Linear, second-order ODEs:

• Homogeneous, linear, second-order ODEs

Week 4

Linear, second-order ODEs:

• Inhomogeneous, linear, second-order ODEs (Method of undetermined coefficients and Variation of parameters)

Week 5

Linear, second-order ODEs:

• Beating, resonance, and damping
• Euler equations

Week 6

Systems of first-order ODEs:

• Homogeneous systems of linear, first-order ODEs

Week 7

Systems of first-order ODEs:

• Inhomogeneous systems of linear, first-order ODEs
• The Lorenz equations, or, the most famous system of ODEs

Week 8

Laplace Transforms:

• Properties of the Laplace transform
• Solving linear ODEs with the Laplace transform

Week 9

Laplace Transforms:

• Step functions and discontinuous forcing
• Impulses
• Convolutions

Week 10

Fourier Series:

• Properties of sine and cosine
• Writing periodic functions as Fourier series

Week 11

Separation of variables for partial differential equations (PDEs):

• Heat equation for a conducting rod with homogeneous boundary conditions
• Heat equation for a conducting rod with inhomogeneous boundary conditions
• Similarity solutions for the heat equation

Week 12

Separation of variables for partial differential equations (PDEs):

• Wave equation for an elastic string
• Propagation of waves on an infinite elastic string
• Laplace equation
• Potential flow around a cylinder

Week 13

Review

Week 14

Final Exam

### Evaluation:

10-20% Quizzes and Homework Midterm Exam(s) Final Exam

### Instructors

Arman Ahmadieh B.Sc., M.Sc. (Sharif University of Technology)
Hayri Ardal, B.Sc.(Bogazici), Ph.D.(Simon Fraser)
Kim Peu Chew, B.Sc. (Nanjing), M.A., Ph.D. (British Columbia)