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

This unit shows how partial differential equations can be used to model phenomena such as waves and heat transfer. The prerequisite requirements to gain full advantage from this unit are an understanding of ordinary differential equations and basic familiarity with partial differential equations.

This unit is an adapted extract from the course *Mathematical methods and models* (MST209

The content acknowledged below is Proprietary (see terms and conditions) and is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence

All materials included in this unit are derived from content originated at the Open University.

1 First-order differential equations

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.

Click 'View document' to open the workbook (PDF, 1.6 MB).

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 *

*All materials included in this unit are derived from content originated at the Open University.*

*Exercise 1*

A vector **a** has magnitude
|**a**| = 7 and direction
*θ* = −70°.
Calculate the component form of **a**, giving the components
correct to two decimal places.

*<*

Author(s):

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Copyright 2009 University of Nottingham

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.

## Velocity and speed

- Author(s):

1.4.2 Displacements and bearings

The **displacement** from a point *P* to a point
*Q* is the change of position between the two points, as
described by the displacement vector

If *P* and *Q* represent places on the
ground, then it is natural to use a bearing to describe the
direct

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* (

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.

3 Work on your own mathematics

Two activities are given below. You are asked to work on them in turn and to record not only your working, but observations on what you notice about your emotions as you work through step by step.

## Activity 3 Constrained numbers

W

This unit focuses on your initial encounters with research. It invites you to think about how perceptions of mathematics have influenced you in your prior learning, your teaching and the attitudes of learners.

The content acknowledged below is Proprietary (see made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence and conditions) and is

All other materials included in this unit are derived from content originated at the Open University.

1. Joi

In this unit you have been introduced to the difference between mathematical content and processes. You have worked on the *do–talk–record* (DTR) framework for learning mathematics.

The *do–talk–record* triad (DTR) is a description of what is likely to take place in collaborative mathematics classrooms. It is concerned with observable events, and with the learner rather than the teacher, though many teaching insights flow from it. Although the order of the triad suggests that it should be followed in a particular sequence, this is not necessarily the case. Sometimes talking comes before doing or recording before talking. It also takes time for a learner to move

## Task 10 The Möbius band

Take a long thin strip of paper (preferably squared or graph paper) about 30 cm by 3 cm. Give one end a half twist and then tape it together. This is a Möbius band as shown in