Course
Discontinued
No
Course Code
MATH 2421
Descriptive
Introduction to Differential Equations
Department
Mathematics
Faculty
Science & 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 ÁñÁ«ÊÓƵ 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.