Course
Discontinued
No
Course code
MATH 2421
Descriptive
Introduction to Differential Equations
Department
Mathematics
Faculty
Science and Technology
Credits
3.00
Start date
End term
201430
PLAR
No
Semester length
15
Max class size
35
Contact hours
4 hours lecture + 1 hour tutorial
Method(s) of instruction
Lecture
Tutorial
Learning activities
Lectures, problems sessions, assignments (written and/or Maple).
Course description
This is a first course in the theory of 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, Euler methods and applications.
Course content
- First-Order Differential Equations: separable, homogeneous, exact, linear, Bernoulli, and Ricatti equations, and applications.
- Higher Order Differential Equations: reduction of order, homogeneous linear equations with constant coefficients, differential operators and undetermined coefficients, variation of parameters
- Equations with Variable Coefficients: Cauchy-Euler equations and power series solutions about ordinary and singular points, Bessel and Legendre Equations
- Laplace Transforms and applications
- Systems of Linear Differential Equations: operator and Laplace transform techniques, systems of first-order equations, reduction of higher-order equations to linear normal form
- Non-linear Systems and Stability: solutions and trajectories of autonomous systems, stability of critical points
- Numerical Solutions: Euler methods
Learning outcomes
Upon completion of MATH 2421 the student should be able to:
- recognise and solve separable, homogeneous, exact and linear first-order differential equations
- determine whether or not a unique solution to a first-order or linear nth-order initial-value problem exists
- solve Bernoulli and Ricatti equations
- determine orthogonal trajectories of a given family of curves
- solve problems involving applications of linear equations including: growth and decay, series circuits, thermodynamics and mixture applications
- solve problems involving applications of non-linear equations including: logistic function, chemical reaction and law of mass action applications
- determine whether or not a set of functions is linearly dependent or independent
- 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
- 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
- analyse problems involving simple harmonic motion
- recognise and solve Cauchy-Euler equations
- use power series techniques to solve differential equations in the neighbourhood of ordinary points
- use the method of Frobenius to solve differential equations about regular singular points
- 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
- reduce higher-order linear differential equations to systems in normal form
- use Euler methods to approximate solutions to differential equations
- analyse trajectories of autonomous first-order differential equations and comment on the stability of critical points
- find equilibrium solutions of second-order differential equations
- find trajectories associated with simple linear and non-linear systems of equations and determine critical points
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% |
Note: All sections of a course with a common final examination will have the same weight given to that examination.
Textbook materials
Textbooks and Materials to be Purchased by Students
Zill, Dennis G., A First Course in Differential Equations with Modeling Applications, 8th Edition, Brooks-Cole, 2005.