Pages in this set

Page 1

Preview of 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

Page 2

Preview of 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

Page 3

Preview of 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…

Page 4

Preview of 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

Page 5

Preview of page 5
Differential Equations Page 5

Page 6

Preview of 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

Page 7

Preview of 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

Page 8

Preview of page 8
nth roots of unity
18 March 2014 12:27




Complex Numbers Page 3

Page 9

Preview of 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

Page 10

Preview of page 10
Expansion of cos(n) and as powers of an
Use:
for
for




Complex Numbers Page 5

Comments

No comments have yet been made

Similar Further Maths resources:

See all Further Maths resources »See all resources »