After studying this unit you should:

• be able to solve homogeneous second-order equations;

• know a general method for constructing solutions to inhomogeneous linear constant-coefficient second-order equations;

• know about initial and boundary conditions to obtain particular values of constants in the general solution of second-order differential equations.

Author(s): The Open University

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): The Open University

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.

Author(s): The Open University

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
Author(s): The Open University

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.

Author(s): The Open University

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
Author(s): The Open University

Acknowledgements

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

Author(s): The Open University

1.5 Exercises

## Exercise 1

A vector a has magnitude |a|Â =Â 7 and direction Î¸Â =Â âˆ’70Â°. Calculate the component form of a, giving the components correct to two decimal places.

<
Author(s): The Open University

Another vector quantity which crops up frequently in applied mathematics is velocity. In everyday English, the words â€˜speedâ€™ and â€˜velocityâ€™ mean much the same as each other, but in scientific parlance there is a significant difference between them.

## Velocity and speed

Author(s): The Open University

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 â€˜15Â° to the East of due Northâ€™, or NÂ 15Â°Â E. This is an instance of a bearing. Directions on the ground are typically given like this, in terms of the directions NorthÂ (N), SouthÂ (S), EastÂ (E)
Author(s): The Open University

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.

Author(s): The Open University

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 aÂ =Â a 1 iÂ +Â a 2 j, what are its magnitude |a| and direction Î¸?

The first part of this question is dealt with using Pythagorasâ€™ Theorem: the magnitude of a v
Author(s): The Open University

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 Î¸ measured anticlockwise from the positive x-axis, then cosÂ Î¸ is the x-coordinate of P and sinÂ Î¸ is the y-coordinate of P (
Author(s): The Open University

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.

Author(s): The Open University

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 SectionÂ 2, in which you will also work with bearings.

This unit is an adapted extract from the Open University course
Author(s): The Open University

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

Author(s): The Open University

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.

Author(s): The Open University

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
Author(s): The Open University

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).

If you find you have a long unmanageable list of favourites/bookmarks you might like to try social bookmarks as an alternative.

## Activity â€“ what you need to know about social bookmarks

Read 7 things you should know about soci
Author(s): The Open University