# Further Pure 2 notes

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- Created by: Kathryn Stephens
- Created on: 26-04-14 14:28

Further Pure 2 notesPDF Document 2.51 Mb

## Pages in this set

### Page 1

First Order Differential equations

03 February 2014 18:46

General Form

Method

Integrating factor =

Multiply throughout by the integrating factor

y

Example

y

y

y

y

Differential Equations Page 1

03 February 2014 18:46

General Form

Method

Integrating factor =

Multiply throughout by the integrating factor

y

Example

y

y

y

y

Differential Equations Page 1

### Page 2

Second Order Differential Equation

02 March 2014 19:55

Form :

Method :

Substitute into the equation

Auxiliary Equation

Cases :

I. +

II. +

III.

Example

When

Solution:

Auxiliary Equation:

or

Case 1

General Solution :

Form :

f(x) = polynomial of degree n

Differential Equations Page 2

02 March 2014 19:55

Form :

Method :

Substitute into the equation

Auxiliary Equation

Cases :

I. +

II. +

III.

Example

When

Solution:

Auxiliary Equation:

or

Case 1

General Solution :

Form :

f(x) = polynomial of degree n

Differential Equations Page 2

### Page 3

Substitute:

into the initial equation

Equate coefficients

f(x) = exponential function of the form

NB if h(x) is part of the auxiliary equation substituting it will give zero, instead use

Substitute:

into the initial equation

Equate coefficients

f(x) = trigonometric function of the form

Substitute:

into the initial equation

Equate…

into the initial equation

Equate coefficients

f(x) = exponential function of the form

NB if h(x) is part of the auxiliary equation substituting it will give zero, instead use

Substitute:

into the initial equation

Equate coefficients

f(x) = trigonometric function of the form

Substitute:

into the initial equation

Equate…

### Page 4

Substitution

17 March 2014 22:23

Example: This equation can't be solved

straight away because it contains

x and y coefficients

1

The substitution is made with e as it

does not change with differentiation

2

3

Sub 2 &3 into 1

Differential Equations Page 4

17 March 2014 22:23

Example: This equation can't be solved

straight away because it contains

x and y coefficients

1

The substitution is made with e as it

does not change with differentiation

2

3

Sub 2 &3 into 1

Differential Equations Page 4

### Page 5

Differential Equations Page 5

### Page 6

Modulus Argument Form

17 March 2014 23:02

Im

Where

z can be written in the form

Re z=r

Where

and is dependant on which quadrant z lies in

Complex Numbers Page 1

17 March 2014 23:02

Im

Where

z can be written in the form

Re z=r

Where

and is dependant on which quadrant z lies in

Complex Numbers Page 1

### Page 7

De Moivre's Theorem

17 March 2014 23:07

Theorem:

Euler's Relation

Proof for De Moivre's Theorem by Mathematical Induction

Cases

I.

=

=

II.

(I)

Cosine is an even function

Sine is an odd function

I, II

Complex Numbers Page 2

17 March 2014 23:07

Theorem:

Euler's Relation

Proof for De Moivre's Theorem by Mathematical Induction

Cases

I.

=

=

II.

(I)

Cosine is an even function

Sine is an odd function

I, II

Complex Numbers Page 2

### Page 8

nth roots of unity

18 March 2014 12:27

Complex Numbers Page 3

18 March 2014 12:27

Complex Numbers Page 3

### Page 9

Trigonometric Identities

18 March 2014 12:32

Expression for and in terms of cosine and since of multiples of

Method:

Group terms and use

Group terms and use

Expansion of cos(n) and as powers of an

Complex Numbers Page 4

18 March 2014 12:32

Expression for and in terms of cosine and since of multiples of

Method:

Group terms and use

Group terms and use

Expansion of cos(n) and as powers of an

Complex Numbers Page 4

### Page 10

Expansion of cos(n) and as powers of an

Use:

for

for

Complex Numbers Page 5

Use:

for

for

Complex Numbers Page 5

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Teacher recommended

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# Further Pure 2 notes

3.0 / 5

- Created by: Kathryn Stephens
- Created on: 26-04-14 14:28

Further Pure 2 notesPDF Document 2.51 Mb

## Pages in this set

### Page 1

First Order Differential equations

03 February 2014 18:46

General Form

Method

Integrating factor =

Multiply throughout by the integrating factor

y

Example

y

y

y

y

Differential Equations Page 1

03 February 2014 18:46

General Form

Method

Integrating factor =

Multiply throughout by the integrating factor

y

Example

y

y

y

y

Differential Equations Page 1

### Page 2

Second Order Differential Equation

02 March 2014 19:55

Form :

Method :

Substitute into the equation

Auxiliary Equation

Cases :

I. +

II. +

III.

Example

When

Solution:

Auxiliary Equation:

or

Case 1

General Solution :

Form :

f(x) = polynomial of degree n

Differential Equations Page 2

02 March 2014 19:55

Form :

Method :

Substitute into the equation

Auxiliary Equation

Cases :

I. +

II. +

III.

Example

When

Solution:

Auxiliary Equation:

or

Case 1

General Solution :

Form :

f(x) = polynomial of degree n

Differential Equations Page 2

### Page 3

Substitute:

into the initial equation

Equate coefficients

f(x) = exponential function of the form

NB if h(x) is part of the auxiliary equation substituting it will give zero, instead use

Substitute:

into the initial equation

Equate coefficients

f(x) = trigonometric function of the form

Substitute:

into the initial equation

Equate…

into the initial equation

Equate coefficients

f(x) = exponential function of the form

NB if h(x) is part of the auxiliary equation substituting it will give zero, instead use

Substitute:

into the initial equation

Equate coefficients

f(x) = trigonometric function of the form

Substitute:

into the initial equation

Equate…

### Page 4

Substitution

17 March 2014 22:23

Example: This equation can't be solved

straight away because it contains

x and y coefficients

1

The substitution is made with e as it

does not change with differentiation

2

3

Sub 2 &3 into 1

Differential Equations Page 4

17 March 2014 22:23

Example: This equation can't be solved

straight away because it contains

x and y coefficients

1

The substitution is made with e as it

does not change with differentiation

2

3

Sub 2 &3 into 1

Differential Equations Page 4

### Page 5

Differential Equations Page 5

### Page 6

Modulus Argument Form

17 March 2014 23:02

Im

Where

z can be written in the form

Re z=r

Where

and is dependant on which quadrant z lies in

Complex Numbers Page 1

17 March 2014 23:02

Im

Where

z can be written in the form

Re z=r

Where

and is dependant on which quadrant z lies in

Complex Numbers Page 1

### Page 7

De Moivre's Theorem

17 March 2014 23:07

Theorem:

Euler's Relation

Proof for De Moivre's Theorem by Mathematical Induction

Cases

I.

=

=

II.

(I)

Cosine is an even function

Sine is an odd function

I, II

Complex Numbers Page 2

17 March 2014 23:07

Theorem:

Euler's Relation

Proof for De Moivre's Theorem by Mathematical Induction

Cases

I.

=

=

II.

(I)

Cosine is an even function

Sine is an odd function

I, II

Complex Numbers Page 2

### Page 8

nth roots of unity

18 March 2014 12:27

Complex Numbers Page 3

18 March 2014 12:27

Complex Numbers Page 3

### Page 9

Trigonometric Identities

18 March 2014 12:32

Expression for and in terms of cosine and since of multiples of

Method:

Group terms and use

Group terms and use

Expansion of cos(n) and as powers of an

Complex Numbers Page 4

18 March 2014 12:32

Expression for and in terms of cosine and since of multiples of

Method:

Group terms and use

Group terms and use

Expansion of cos(n) and as powers of an

Complex Numbers Page 4

### Page 10

Expansion of cos(n) and as powers of an

Use:

for

for

Complex Numbers Page 5

Use:

for

for

Complex Numbers Page 5

## Comments

No comments have yet been made

## Similar Further Maths resources:

0.0 / 5

0.0 / 5

5.0 / 5

Teacher recommended

0.0 / 5

0.0 / 5

0.0 / 5

5.0 / 5

## Related discussions on The Student Room

- AQA A2 Maths (Old Spec) - Further Pure 2 MFP2 - 22 June ... »
- Question about OCR FP3 Further Pure 2+3 Book by Quadling ... »
- edexcel igcse further pure mathematics 12th june 2018 »
- AQA AS Maths (Old Spec) - Further Pure 1 MFP1 - 13 June ... »
- Edexcel AS Mathematics: Further Pure FP1 6667 01 - 14 May ... »
- Edexcel A2 Mathematics: Further Pure FP2 6668 01 - 06 June ... »
- AQA A2 Maths 2017 - MFP4 Further Pure 4 - Wednesday 24 ... »
- Edexcel Mathematics: Further Pure FP2 6668 01 - 07 Jun ... »
- Edexcel Mathematics: Further Pure FP1 6667 01 - 19 May ... »
- Edexcel A2 Mathematics: Further Pure FP3 6669 01 - 25 June ... »

## Comments

No comments have yet been made