Introduction to Differential Equations

Curriculum guideline

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Course
MATH 2421
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
  1. First-Order Differential Equations:  separable, homogeneous, exact, linear, Bernoulli, and Ricatti equations, and applications.
  2. Higher Order Differential Equations:  reduction of order, homogeneous linear equations with constant coefficients, differential operators 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:  operator and Laplace transform techniques, systems of first-order equations, reduction of higher-order equations to linear normal form
  6. Non-linear Systems and Stability:  solutions and trajectories of autonomous systems, stability of critical points
  7. 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.

Prerequisites

MATH 1220 and MATH 2232 or instructor permission