Differential Equation Modelling & Solution (U01462)

? Credit Points : 10 ? SCQF Level : 8 ? Acronym : MAT-1-DMS

First year course, primarily for Honours Degrees in Mathematics and/or Statistics. Syllabus summary: Rate of change as a derivative. Velocity, acceleration, Newton's second law, including resistance. Modelling of population dynamics. Analytical methods for solving separable ordinary differential equations (ODEs) and linear first order ODEs with constant coefficients. Numerical and graphical methods of solution (including use of Maple). Truncation error and order. The idea of equilibrium. The topics will be developed through examples and practical questions, such as "Why do small hailstones not kill you?"; the motion of falling bodies; conservation of fish stocks: the effects of harvesting and overcrowding; Lotka-Volterra two-component models.

Entry Requirements

? Pre-requisites : H-Grade Mathematics or equivalent

Subject Areas

Home subject area

Specialist Mathematics & Statistics (Year 1), (School of Mathematics, Schedule P)

Delivery Information

? Normal year taken : 1st year

? Delivery Period : Semester 2 (Blocks 3-4)

? Contact Teaching Time : 2 hour(s) 30 minutes per week for 11 weeks

All of the following classes

Type	Day	Start	End	Area
Lecture	Monday	10:00	10:50	Central
Lecture	Wednesday	10:00	10:50	Central

? Additional Class Information : Tutorials: F at 0900, 1000 and 1110.

Summary of Intended Learning Outcomes

1. Familiarity with Newton's second law for linear motion under gravity, with linear or quadratic resistance, or with buoyancy: and familiarity with formulating one-dimensional dynamics as an O.D.E. with initial condition(s).
2. Familiarity with the notions of dependent and independent variable, and order of a differential equation (up to second).
3. Ability to recognise and solve separable equations (analytically).
4. Ability to recognise and solve (analytically) linear first-order equations with constant coefficient and right-hand side equal to zero, constant, exponential, sine or cosine. Solution of y' = ky to be written at sight.
5. Ability to sketch direction fields and solution in simple cases (y' a function of x or of y only).
6. Ability to solve and plot solutions numerically (using Maple). Understanding of the concepts of truncation error (local and global), and order. Ability to apply Euler's method for one or two steps.
7. Understanding of the basic models of population dynamics: Malthus, constant harvesting, logistic; their solution, interpretation and comparison with experimental evidence.
8. Formulating and solving simple compartment models (e.g. mixing).
9. Knowledge of equivalence of system of two first order equations to a given second order equation. Finding phase plane equation for autonomous system of two first-order equations, and an integral.
10. Familiarity with the qualitative behaviour of the solutions of the Lotka-Volterra system, including its equilibrium and a first integral.
11. Ability to recognise the equation of displaced simple harmonic motion, and to write down the solution of the SHM equation at sight. Knowledge of its period.