Further Pure 3
Chapter 1  Series and Limits (18.1)


1.1 The concept of a limit 1.2 Finding limits in simple cases 1.3 Maclaurin’s series expansion 1.4 Range of validity of a series expansion 1.5 The basic series expansions 1.6 Use of series expansions to find limits 1.7 Two important limits 1.8 Improper integrals 
Use of the range of values of x for which these expansions are valid, as given in the formulae booklet, is expected to determine the range of values for which expansions of related functions are valid;
In this chapter, it is shown how series expansions are used to find limits and how improper integrals are evaluated. When you have completed it you will:

Chapter 2  Polar Relationships (18.2)


2.1 Cartesian and polar frames of reference 2.2 Restrictions on the values of θ 2.3 The relationship between Cartesian and polar coordinates 2.4 Representation of curves in polar form 2.5 Curve sketching 2.6 The area bounded by a polar curve 
This chapter introduces polar coordinates. When you have completed it, you will:

Chapter 3  Differential Equations (18.3)


3.1 The concept of a differential equation: order and linearity 3.2 Families of solutions, general solutions and particular solutions 3.3 Analytical solution of first order linear differential equations: integrating factors 3.4 Complementary functions and particular integrals 3.5 Transformations of nonlinear differential equations to linear form 
The relationship of order to the number of arbitrary constants in the general solution will be expected. This is the first of three chapters on differential equations. When you have completed it you will:

Chapter 4  Differential Equations – 1^{st} Order
(18.4) 
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4.1 Introduction 4.2 Euler’s formula 4.3 The midpoint formula 4.4 The improved Euler formula 4.5 Error analysis: some practical considerations 
To include use of an integrating factor and solution by complementary function and particular integral. Formulae to be used will be stated explicitly in questions, but candidates should be familiar with standard notation such as used in: This chapter gives an introduction to numerical methods for solving first order differential equations. When you have completed it, you will:

Chapter 5  Differential Equations – 2^{nd} Order
(18.5) 
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5.1 Introduction to complex numbers 5.2 Working with complex numbers 5.3 Euler’s identity 5.4 Formation of second order differential equations 5.5 Differential equations of the form 5.6 Differential equations of the form 5.7 Solution of second order differential equations by reduction to simultaneous first order differential equations 
Including repeated roots. Finding particular integrals will be restricted to cases where f (x) is of the form ekx , cos kx , sin kx or a polynomial of degree at most 4, or a linear combination of any of the above. Level of difficulty as indicated by: (a) Given x^{2}d^{2}y/dx^{2} – 2y = x use the substitution x = e^{t} to show that d^{2}y/dt^{2} – dy/dt – 2y = e^{t} Hence find y in terms of t Hence find y in terms of x (b) Use the substitution u = dy/dx to show that and hence that where A is an arbitrary constant. Hence find y in terms of x. This chapter deals with analytical methods for solving second order linear differential equations with constant coefficients. When you have completed it, you will:
