Introduction This unit extends the ideas introduced in the unit on first-order differential equations to a particular type of second-order differential equation which has a variety of applications. The unit assumes that you have previously had a basic grounding in calculus, know something about first-order differential equations and have some familiarity with complex numbers. This unit is an adapted extract from the course Author(s):
Acknowledgements The content acknowledged below is Proprietary (see terms and conditions) and is used under licence. All materials included in this unit are derived from content originated at the Open University.
Learning outcomes After studying this unit you should be able to: understand how the wave and diffusion partial differential equations can be used to model certain systems; determine appropriate simple boundary and initial conditions for such models; find families of solutions for the wave equation, damped wave equation, diffusion equation and similar homogeneous linear second-order partial differential equations, subject to simple boundary conditions, using the meth
Learning outcomes During this unit you will: learn some basic definitions and terminology associated with differential equations and their solutions; be able to visualize the direction field associated with a first-order differential equation and be able to use a numerical method of solution known as Euler's method; be able to use analytical methods of solution by direct integration; separation of variables; and the integrating factor method.
First-order differential equations This unit introduces the topic of differential equations. The subject is developed without assuming that you have come across it before, but it is taken for granted that you have a basic grounding in calculus. In particular, you will need to have a good grasp of the basic rules for differentiation and integration. This unit is an adapted extract from the course Mathematical methods and
Acknowledgements All materials included in this unit are derived from content originated at the Open University.
1.5 Exercises A vector a has magnitude
|a|Â =Â 7 and direction
θ = −70°.
Calculate the component form of a, giving the components
correct to two decimal places. 1.4.3 Velocity Another vector quantity which crops up frequently in
applied mathematics is velocity. In everyday English, the
words ‘speed’ and
‘velocity’ mean much the same as each other,
but in scientific parlance there is a significant difference between
them. 1.4.1 Bearings In the following subsections, we apply the vector ideas
introduced so far to displacements and velocities. The
examples will feature directions referred to points of the compass,
known as bearings. The direction of Leeds relative to Bristol can be described
as ‘15° to the East of due North’,
or N 15° E. This is an instance of a
bearing. Directions on the ground are typically given like
this, in terms of the directions North (N),
South (S), East (E) 1.3: Summing vectors given in geometric form The following activity illustrates how the conversion
processes outlined in the preceding sections may come in useful. If
two vectors are given in geometric form, and their sum is sought in
the same form, one approach is to convert each of the vectors into
component form, add their corresponding components, and then convert
the sum back to geometric form. 1.2: Converting to geometric form You have seen how any vector given in geometric form, in
terms of magnitude and direction, can be written in component form.
You will now see how conversion in the opposite sense may be
achieved, starting from component form. In other words, given a
vector
a = a
1
i + a
2
j,
what are its magnitude |a| and direction θ? The first part of this question is dealt with using
Pythagoras’ Theorem: the magnitude of a v 1.1: Converting to component form In some applications of vectors there is a need to move
backwards and forwards between geometric form and component form; we
deal here with how to achieve this. To start with, we recall definitions of cosine and sine. If
P is a point on the unit circle, and the line segment
OP makes an angle θ measured anticlockwise
from the positive x-axis, then
cos θ is the x-coordinate of
P and sin θ is the
y-coordinate of P ( Learning outcomes On completion of this unit you should be able to: convert a vector from geometric form (in terms of
magnitude and direction) to component form; convert a vector from component form to geometric form; understand the use of bearings to describe direction; understand the difference between velocity and speed; find resultant displacements and velocities in geometric
form, via the use of components. Introduction In this unit you will see first how to convert vectors from geometric form, in terms of a magnitude and direction, to component form, and then how conversion in the opposite sense is accomplished. The ability to convert between these different forms of a vector is useful in certain problems involving displacement and velocity, as shown in Section 2, in which you will also work with bearings. This unit is an adapted extract from the Open University course Acknowledgements All materials included in this unit are derived from content originated at the Open University. 1.3 Designing alternative programmes and curricula Assuming that both the content of mathematics and the processes need to be included in programmes and curricula, the problem becomes one of how a suitable curriculum can be structured. One possibility is to construct a very specific curriculum with clearly defined objectives for both content and processes separately, and possibly with suggested learning activities. However, content and process are two complementary ways of viewing the subject. An alternative is to see the curriculum in Learning outcomes After studying this unit, you should: understand some current issues in mathematics education, such as the relationship of mathematics content to mathematics processes.
understand a variety of approaches to the teaching of mathematics such as 'do-talk-record' be able to approach mathematical problems and tasks in a flexible way. Introduction This unit is aimed at teachers who wish to review how they go about the practice of teaching maths, those who are considering becoming maths teachers, or those who are studying maths courses and would like to understand more about the teaching process. This unit is from our archive and is an adapted extract from Teaching mathematical thinking at Key Stage 3 (ME624) which is no longer taught by The Open University. If you want to study formally with us, you may wish to explore other cour 1.1 Workbook contents The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. Section 4.2 of the unit requires you to listen to some audio files. You'll find these on the next page of this unit. Click on 'View document' to open the workbook (PDF, 4 MB). 1.5.5 Social bookmarks If you find you have a long unmanageable list of favourites/bookmarks you might like to try social bookmarks as an alternative. Read 7 things you should know about soci
Exercise 1
Velocity and speed
Author(s):
Activity – what you need to know about social bookmarks
