Book: Differential
Equations and Boundary Value Problems:
Computing and Modeling, 2nd Edition, by Edwards & Penney.
What are Differential Equations?
The study of Differential Equations owes its origins to problems in classical Mechanics from the 17th century, but today the applications have spread to disciplines such as Chemistry, Biology, Economics and Finance. To give a simple example of a differential equation, let us take an example from Mechanics, a spring-mass system.
--\ Spring Constant k <--- Force = -k x
|| /\ /\ /\ /\ |-------|
=======/ \ / \ / \ / \ =======| |
| \/ \/ \/ \/ | Mass |
| | m |
--/ |-------| |<-- x -->|
| |
Equilibrium Displacement from Equilibrium
The displacement of the body from the equilibrium position where the spring is relaxed is measured by 'x'. Hooke's Law states for small displacements 'x', the force exerted by the spring on the body is proportional to 'x' and in the opposite direction. Newton's Law, F = ma, implies that if we ignore friction and the mass of the spring then the mass times the acceleration of the body is equal to -kx. Since the acceleration of the body is equal to the second derivative of 'x' with respect to time 't' this gives us the equation:
d^2 x(S) m ----- = -k x.
d t^2 The equation (S) is an equation in which 'x' is implicitly a function of the independent variable 't'. Because (S) is an equation involving an unknown function 'x(t)' and the derivatives of 'x(t)' with respect to the independent variable 't' we call (S) a differential equation.
What do I expect you to learn in this course?
In this course we will study differential equations like (S). By the end of the course you should have a thorough understanding of how to solve most of the common linear, scalar differential equations that are encountered in practice; you should be able to solve systems of linear differential equations with constant coefficients; you should have an understanding of the phase portraits of constant coefficient linear 2-dimensional systems of differential equations; you will get an understanding of how computers numerically calculate (approximate) solutions to differential equations and you will get to see how many 'classical' functions like 'exp', 'sin', 'cos' and more arise as solutions of 'classical' differential equations.
How you will be evaluated:
Grading:
Maple 5
Quizzes 25
Midterm 30
Final Exam 40
Total 100
This grading scheme is tentative and changes may be made. All changes will be announced well in advance of the final exam.
The class will consist of both lecture and discussion. There will be a short quiz every week after a question and answer session devoted to understanding the material better. There will be homework problems assigned every week also. Students are encouraged to do the homework as working on these problems is the best way to learn the material and do well in the class, but however, these homework assignments will not be graded and will not count towards the final grade
The final exam will cover the entire course.