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MAT 1092
Mathematical Methods
for
ICT
Dr Matthew Montebello
CB 409
[email protected]
CSA 3210 Agent
Technology
CSA 3210
Agent Technology
Lecture Schedule
1.
2.
3.
4.
5.
6.
7.
Introductions & Matrices
Determinants
Solution of Linear Equations
Eigenvalues and Diagonalisation
Ord. Diff. Eqns. of 1st Order
Ord. Diff. Eqns. of 2nd Order + constant coeffs.
Partial Diff. and Exact Diff. Eqns.
MAT 1092
Mathematical Methods for ICT
Useful details
• Assessment – 100% exam
• Exam type examples and exercises will be given
during/at-the-end of each lecture/topic
• Notes are online at:
http://staff.um.edu.mt/mmon1/lectures/mat1092
• Past-papers are available online
• Feel free and comfortable to:
– Ask questions
– Sort out queries right away
• Best practice is to:
– Work out examples and understand every step
– Not to push back unclear issues
MAT 1092
Mathematical Methods for ICT
Books
• A Course in Pure Mathematics,
– Gow M., Arnold, 1960
• Elementary Linear Algebra,
– Andvilli S. and Hecker D., Harcourt Acad. Press, 1999
• Advanced Mathematics,
– Kreysig E., John Wiley & Sons, 8th edition, 1998
• Elementary Differential Equations,
– Derrick W. and Grossman S., Addison-Wesley, 4th
edition, 1997.
MAT 1092
Mathematical Methods for ICT
1. Introduction
a.
b.
c.
d.
Welcome – Introductions
Run through the schedule
Useful details & Books
Preliminary stuff:
–
–
–
–
–
–
What are Matrices?
Determinants!!! In relation to matrices?
Do you remember Linear Equations?
Fancy an Eigenvector? What?
Differential equations of the order of ???
What is the difference between partial and exact?
MAT 1092
Mathematical Methods for ICT
1.1 What are Matrices?
Wikipedia gives a good definition, namely:
“a rectangular table of numbers”
These rectangular shaped groups of numbers can be:
• Added
• Subtracted
• Multiplied
• Divided?
So what is an:
• Identity matrix
• Inverse matrix
MAT 1092
Mathematical Methods for ICT
1.2 Matrix Addition and Subtraction
Matrix addition/subtraction is performed by
adding/subtracting the individual entries with the
identical row and column position of the two
matrices.
This means that the two matrices have to necessarily
be of the same size in order to perform these
operations.
Hence, the sum/difference A+/-B of two n x m
matrices results in an n x m matrix having Ai j + Bi j
as its i-j entry.
E.g.
1 2 -3
-1 6 3
0 8 0
4 0 2 +
8 12 14 = 12 12 16
5 8 0
2 1 0
7 9 0
1 2 -3
4 0 8
5 8 0
MAT 1092
-
-1 6 3
8 12 4 =
2 1 0
Mathematical Methods for ICT
2 -4 -6
-4 -12 4
3 7 0
1.3 Matrix Multiplication
Matrix multiplication is performed by multiplying
each entry from every row of the first matrix with
each entry of every column in the second matrix,
and adding up the partial sums.
This means that the width of the first matrix has to be
equal to the height of the second matrix.
If A is an n x h matrix and B is an h x m matrix then
the product AB, is defined to be an n x m matrix
with:
(AB)i j = Ai1 B1j + Ai2 B2j + … + Ai h Bh j
E.g.
1 6 3
4 1 2 . 8 2 4
5 -3 0 2 1 0
MAT 1092
16 28 16
=
-19 24 3
Mathematical Methods for ICT
1.4 Matrix Multiples
To work out the multiples of a matrix is simply
multiplying each entry within the matrix with the
multiple value.
So, if the matrix A was to be multiplied by a value n,
then the resultant matrix will have every entry
multiplied by n.
That is the i-j entry in matrix A will become n(i-j).
E.g. 6 . 2 1 3 = 12 6 18
1 4 2
6 24 12
The identity matrix of a square matrix is
1 0 0
0 1 0
0 0 1
while the product of a matrix and its’ inverse is equal
to the identity matrix.
MAT 1092
Mathematical Methods for ICT
1.5 Matrices - Check Point
If A = 0
4
5
work out:
-1
0
-1
2
1
3
MAT 1092
B = -1 2
0 3
4 -2
0
1
0
a)
A+B
b)
AxB
c)
2A
d)
3B
e)
2A - 3B
f)
2A x B
g)
A x A or A2
h)
5B3
Mathematical Methods for ICT
1.6 Inverse of a Matrix
If A
is the
an (n
x n) square
andmatrices
there is aif
Work
out
inverse
of the matrix
following
matrix B with the property that: A . B = I
possible:
1
3 to be the inverse of A
-2 and
3/2 is denoted A-1
B
is
defined
-1
a. A =
A =
2
4
1
0
AA1  I
1
1
4
-½
A1 A  I
0
-1
If A has an inverse, it is called
invertible.
-1
-2 ½ ½
b. B = 1 2 0
B =
Otherwise,3 it is
called singular. -3 0 1
0 4
How do you work out the inverse of Matrix A?
1
0 0
1
0 0
i.c. Write
down
the
matrix
[A
|
I
]
-1
C = 2 -1 0
C =
2
-1 0
ii. Row-reduce [A | I].
1
1 1
-3 1 1
iii. If the reduced form is [I | B]
Not
-1
3
0
1
d. then
D = A is invertibleDand= B = A-1
possible
2
-1 0
is
else A is singular.
singular 1 1 1
Some Inverse Matrix examples ...
MAT 1092
Mathematical Methods for ICT
1.6 Inverse of a Matrix
If A is an (n x n) square matrix and there is a
matrix B with the property that: A . B = I
B is defined to be the inverse of A and is denoted A-1
AA1  I
If A has an inverse, it is called invertible.
Otherwise, it is called singular.
How do you work out the inverse of Matrix A?
i. Write down the matrix [A | I ]
ii. Row-reduce [A | I].
iii. If the reduced form is [I | B]
then A is invertible and B = A-1
else A is singular.
Some Inverse Matrix examples ...
MAT 1092
Mathematical Methods for ICT
1.7 What is a Determinant?
Determinants
are mathematical
objectsmatrices
that are very
Work out the inverse
of the following
if
useful
in
the
analysis
and
solution
of
linear
possible:
equations.
1
3
=
det(A)
= 0, the matrix is said to
Ifa. theAdeterminant
of
a
matrix
is
2
4
be singular, and if the determinant is 1, the matrix is
0 1
said to be1 unimodular.
b. B
= 1 2 0 of a Matrix
det(B)A=is defined to be:
The
determinant
a
A=
c. C c=
a
d
d. D g=
3
0
4
1
0
0
2
-1 0
b
d
b1 c 1
det(A) = |A| = ad – bc
det(C) =
1
 e
e f
det(A)  a det 
3
0 1
 h

h i
2
-1 0
 d
f 

 b det 


det(D) i=  
 g
 d e  
f 


 c det 




i 
  g h 
1
1 1
Work out determinants
of previous matrices ...
MAT 1092
Mathematical Methods for ICT
1.8 Linear Equations
Also referred to as Straight Line graphs, linear
equations are of the form:
40
30
20
y = mx +c
10
Where
m is known as the gradient or slope, and
c is known as the intercept
0
-10
-20
The standard form is ax
+ by = c
-30
e.g. Solve y = 18 – 4x and y = 9x + 5
1 18+ y = 5
or 4x + y = 18 4 -9x
1
R1/4
-9
MAT 1092
1
¼ 4½
-9
5
1
¼
4½
0
3¼ 45 ½
4
0
4
0
1
14
1
5
R2+9R1
4R1-R2
1
¼
4½
0
1
14
x
=
1
1
0
1
0
1
14
R2/3¼
R1/4
Mathematical Methods for ICT
y = 14
1.9 Eigenvalues & Eigenvectors
AThe
vector
may
be thought X,
of for
as an
arrow.
set of
eigenvectors,
A is
defined as those
•vectors
It has awhich,
length,when
called
its magnitude
value)
multiplied
by A, (scalar
result in
a
•simple
It points
in some
direction.
scaling
λ ofparticular
X. So bascially
λ is a constant.
A linear transformation may be considered to operate
This
= λ.X
(λ.I).X changing both its
on ameans
vectorthat:
to A.X
change
it, =usually
magnitude and its direction.
 (A – λI) X = 0
An Eigenvector of a given linear transformation is a
is multiplied
constant
called we
the
Invector
orderwhich
for this
equation by
to ahave
solutions
Eigenvalue
during det(A
that − λI),
transformation.
The
require
the determinant
to be zero.
direction of the eigenvector is either unchanged by
that
det transformation
= det 1-λeigenvalues)
=0
3
1 (for
0
1
3
positive
or
- λ
4
2-λ
reversed4 (for
0 eigenvalues).
1
2 negative
Eigenspace corresponding to a given eigenvalue of a
transformation
is2-12
the =vector
space
linear
(1-λ)(2-λ)-12
= 2-3λ+λ
λ2-3λ–10
= 0of all
eigenvectors with that eigenvalue
 (λ – 5) (λ + 2) = 0
1
3
Consider the square matrix A= 1 3
4
2
4
2
 λ = 5 and λ = -2 The Eigenvalues
of A
MAT 1092
Mathematical Methods for ICT
1.10 Differential Equations
Differential equations are simply equations involving
an unknown factor, like x, and its derivatives.
What is an unknown factor and its derivatives?
If it is a function of x: y = f(x) = 3x2 + 4x
Then its differentiation is y’ = f’(x) = 6x + 4
So a function like: f’(x) + 7x – 3 is a Differential Eq.
But what is the exact meaning of Differentiation?
The derivative provides the measure of the slope
(gradient) of a function
-
The gradient of a function f(x) = y is found by
y/x
• Rewritten in terms of the above equation,
slope = f(x) /x
MAT 1092
Mathematical Methods for ICT
1.11 Differential Rules
3). What
•e.g.Constant
If f(x)
=
k,
then
f’(x)
=
0.
supposeRule:
y = ln(x
is
dy/dx?
3x2, what
e.g. Suppose f(x) = e2x
is What
f’(x)?is f’(x)?
/
(x-2).
e.g. Suppose f(x) = 3. What is f’(x)?
Try out
n, then f’(x) = cnxn-1
2
•Try
Power
Rule.
If
f(x)
=
cx
out
out:notorious
a)25x derivatives:
/ 5x
Some
a) Suppose
Whatisisf’(x)?
f’(x)?
e.g.
Supposef(x)
f(x)==ln(x).
3x2, what
If y =b)7e6 /what
is dy/dx?
x Rule:
• a)
Sum-Difference
If
f(x)
=
g(x)
±
h(x)
2
b)
Suppose
f(x)
=
log(3x
+
x
).
What
is
f’(x)?
Trythen
out f’(x)
the following:
2 3+2lnx
3 –e
g’(x)
h’(x)
b) If f(x)c)==(16x
5x
+±12x)
/22x
4 ln(x3)
ln
xRule:
then
dy/dx
1/x
c)y =Suppose
y =If3x.ln(5x
+=2x)
• a)IfProduct
f(x)
=
g(x).h(x)
3
10x out:
Try
f’(x)=g’(x).h(x)+g(x).h’(x)
Ifthen
y = sin
x then dy/dx = cos x
• b) Quotient
If f(x)=g(x)/h(x)
a)6xf(x) = Rule:
17 – 4x
2
3
2
then
f’(x)=[g’(x).h(x)-g(x).h’(x)]
/
[h(x)]
If
y
=
cos
x
then
dy/dx
=
-sin
x
f(x)2 = (4x )(5-x ). What is f’(x)?
c) b)e.g.
2xy2 +Suppose
1
3
=
12x
–
4x
+
5x
–
7
2
e.g.
Suppose f(x) = 2x / (x-2). What is f’(x)?
Try
out:
2x
If
y
=
tan
x
then
dy/dx
=
sec
• Log
If 6x
f(x) = ln( g(x) )
c) yRule:
= 3x2 – 4 +
a) y f’(x)
= 6x(2x
– 5) / g(x)
then
=
g’(x)
x
If y = ex
then
dy/dx
=
e
b) f(x) = (12x3 – 4x2 +Rule:
5x – If
7)(3x
• Exponential-Function
f(x) –=7)
eg(x)t
g(x)
then
f’(x)
= g’(x).e
c) f(x)
= (17
– 4x)(5)
MAT 1092
Mathematical Methods for ICT
1.12 Partial & Exact Diff. Equations
All these differentiations of y = f(x) produced:
y’ = dy/dx
This is called Differential Equation of 1st. Order
What happens if you workout the derivative of a
1st.order differential equation?
You produce a 2nd. Order differential equation.
A first-order differential equation is called exact, or an
exact differential, if it is the result of a simple diff.
If the unknown function depends on more than 1
variable, then it is called a partial differential
MAT 1092
Mathematical Methods for ICT
1.13 Closure - Lecture 1
In this lecture:
• Course details
• Schedule
• Books – WWW
• Basic issues:
–
–
–
–
–
–
–
Matrices +, -, *, /, inverse
Determinants & row reduction
Linear Equations
Eigenvalues & eigenvectors
Differential equations – rules
1st order, 2nd order
Partial and Exact differential equations
All this will be expanded in the following sessions.
MAT 1092
Mathematical Methods for ICT
2. Matrices
a.
b.
c.
d.
e.
f.
g.
h.
i.
Introduction
Multiplication
Gaussian Row Reduction technique
Cramer’s Rule
Inverse – invertible / singular
Diagonizable
Transpose
Matrix Rank
Closure
MAT 1092
Mathematical Methods for ICT
2.1 Matrices Introduction
Before proceeding we need to ensure that the
following operations are understood:
a. Addition
b. Subtraction
c. Multiples
d. Multiplication
e. Inverse – Row Reduction
f. Determinant
MAT 1092
Mathematical Methods for ICT
2.2 Some Exercises (past-papers)
a. Feb 07
i. No.1 – first part
ii. No.3 (a)
iii. No.3 (b) (i)
b. Jan 05
i. No.1 (a)
ii. No.2 – first part
iii. No.3 – first part
c. Jan 06
i. No.1 – first part
ii. No.2 – first part
iii. No.3 (b) – first part
MAT 1092
Mathematical Methods for ICT
2.3 Gaussian Row Reduction
A sequence of elementary row operations is then
applied to this matrix so as to transform it to row
echelon form. The elementary operations are:
• Row Scaling: Multiply a row by a scalar ≠ 0;
• Row Exchange: Exchanges two rows;
• Row Replacement: Add multiple of row to another;
Such operations are referred to as multiplication with
Elementary Matrices, E, that have the same effect.
1
0
0
0
0
1
1
0
0
0
m
0
0
1
0
0
1
0
0
0
1
1
0
0
m
0
1
MAT 1092
Mathematical Methods for ICT
2.4 Cramer’s Rule
• Look at the following system of equations:
z ==33
2x
2 2x+ +1y
1 y ++ 11z
D=
11xx–-1
-1==0 0 Ans Col=
–1y
y – z1z
1x
1 x ++ 2y
22y ++ 11z
z ==00
To find x
To find y
To find z
3 1
1
2
3
1
2
1
3
0 -1 -1
1
0
-1
1
-1 0
0 2
1
1
0
1
1
2
0
Dx
Dy
Dz
Cramer's Rule says that:
x = Dx ÷ D
y = Dy ÷ D
z = Dz ÷ D.
x = 3/3 = 1
y = –6/3 = –2
z = 9/3 = 3
E.g. See Feb 07 no.3(b)(ii) & Jan 05 no.2(b)
Previous Example
MAT 1092
Mathematical Methods for ICT
2.5 Inverse of a Matrix
Recall that the inverse of A and is denoted A-1 where
A.A-1 = I (identity matrix or 1)
We have seen in Slide 1.6 how to determine the
inverse of a matrix.
The reason we work out the inverse of a matrix is
mainly to solve a system of linear equations.
You can write a system in matrix form as AX = B.
Now, pre-multiply both sides by the inverse of A.
A-1.A.X = A-1.B
X = A-1.B
e.g.1
2x + y + z = 3
x–y–z=0
x + 2y + z = 0
MAT 1092
e.g.2
3x + 2y - 5z = 12
x - 3y + 2z = -13
5x - y + 4z = 10
Mathematical Methods for ICT
2.6 Is a Matrix Diagonalizable?
A square matrix A is called diagonalizable if
it is similar to a diagonal matrix.
i.e. if there exists an invertible matrix P such
that:
P−1AP is a diagonal matrix, D
where the eigenvalues of A are the entries of
D and P is an invertible matrix consisting
of the eigenvectors corresponding to the
eigenvalues in P itself.
The main Diagonal Line of a square matrix is:
a
0
0
0
b
0
0
0
c
MAT 1092
Mathematical Methods for ICT
2.7 Transpose of a Matrix
The transpose of a matrix is the matrix formed
by “flipping” about the diagonal line from
the upper left corner.
It is usually denoted as At
If A = a
b
c
Then At =
d
e
a
d
b
e
c
f
MAT 1092
f
Mathematical Methods for ICT
2.8 Rank of a Matrix
The rank of a Matrix is equal to the number of
linearly independent rows or columns of
the matrix.
Check for dependencies between the rows and
columns by:
• Visually inspecting them
• Working out the determinant ≠ 0
The maximum number of rows or column is
the value given to the rank.
e.g.1.
Feb 07 no.2(a)i-iii.
e.g.2.
Jan 06 no.3(a)
MAT 1092
Mathematical Methods for ICT
2.9 Closure – Lecture 2
a.
b.
c.
d.
e.
f.
g.
h.
i.
Introduction
Multiplication
Gaussian Row Reduction technique
Inverse – invertible / singular
Diagonizable
Transpose
Rank
Closure
Tutorial – extra self-inflicted work
MAT 1092
Mathematical Methods for ICT
2.10 Extra Exercises
Typical Exam Question:
• If A is the matrix formed from the lefthand side of the equations:
• While B is the matrix formed 2x – y + 3z = 13
2y – 4x – z = -5
from their answers.
Determine:
3z – 2y = 6
a. A.B
b. At
c. A-1
d. Bt
e. Bt.A-1
f. |A|
g. The values of x, y, and z.
i. Using A-1
ii. Employing Cramer’s Rule
MAT 1092
Mathematical Methods for ICT
2.11 Solution to Previous Question
2x
3z3z
=
1313
22x -–y1y
-1+ +
3=
-4x++2y
1z= =-5-5
-5
A=
B=
-4
22y––z-1
-4x
2y++3z3z
00x––2y
-2
3 ==66
0x
1/5 find
-3/20 z-1/4
2 find
-4 0y (c)A-1= To
To find49x (b)At= To
(a)A.B=
-½
2 -2 3
13 -1-68 1
2 -1 13
2 3/5 -13/10 13
2/5
1/5
0
-5 2 28 -1
-4 3 -1
-5 3 -1
-4 2
-5
-2 6
-3/4
13 -5 3
6 (e) B0t.A-1=6 2 -9/4 3
(d)6Bt= -2
(f)0|A|=20
Dx
Dy
Dz
Cramer's Rule says that:
x = Dx ÷ D
y = Dy ÷ D
z = Dz ÷ D.
x = 37/20
y = 66/20 = 33/10
z = 84/20 = 21/5
MAT 1092
Mathematical Methods for ICT
3. Determinants
1. Introduction
2. Working out the determinant
3. Used to determine
i. Invertible matrices
ii. Linear equations
iii. Eigenvalues
4. Determinant effect on row reduction
5. Worked examples
6. Closure
MAT 1092
Mathematical Methods for ICT
3.1 Determinants Introduction
The determinant is defined as the value
associated with every square matrix that
has particular significant properties to that
same matrix, like:
– Switching two rows or columns changes the sign.
– Scalars can be factored out from rows and
columns.
– Multiples of rows and columns can be added
together without changing the determinant's
value.
– Scalar multiplication of a row by a constant
multiplies the determinant by .
– A determinant with a row or column of zeros has
value 0.
– Any determinant with two rows or columns equal
has value 0.
MAT 1092
Mathematical Methods for ICT
3.2 Determinant Value
How do you work out the determinant?
The determinant of a Matrix A is defined to
be:
A=
A=
a
b
c
d
a
b
c
d
e
f
g
h
i
det(A) = |A| = ad – bc
 e f 
 d f  
 d e  





det(A)  a det 
 b det 
 c det 






h
i
g
i
g
h












det(A) = a(ei – fh) – b(di – fg) + c(dh – ge)
MAT 1092
Mathematical Methods for ICT
3.3 Determinant Uses
• If the determinant of a matrix is
equal to zero THEN the inverse
does not exist;
• The determinant of a matrix is
extensively employed in Cramer’s
rule to work out the solutions of
linear equations;
• It is also employed to workout the
eigenvalues of a matrix. In fact the
product of the eigenvalues is equal
to the determinant of the matrix.
MAT 1092
Mathematical Methods for ICT
3.4 Determinant Effect
Consider the matrix A =
6 12 15
B=
3 -2 0
4
C=
0 -5
2
4
5
3
-2 0
R2  R3
4
0
R3 + R1
3R1
-5
2 4 5
2
4
5
4
0
-5
D=
3 -2 0
3
-2 0
6
4
0
Notice the determinants of all the matrices:
|A| = 120
|B| = 360
|C| = -120
|D| = 120
and appreciate that:
• Row scaling multiplies determinant by scalar
• Row exchange negates the determinant
• Row replacement keeps the same determinant
MAT 1092
Mathematical Methods for ICT
3.5 Determinant Examples
Work out the determinants of the following matrices
Some
more example to work from Past Papers:
if possible:
• Feb 07 1 3
a. A =
det(A) =
• no. 3(a) 2
4
• no. 3(b) (i) & (ii)
•b.JanB05
=
• no.3
1
0
1
1
2
0
3
0
4
• Jan 06
1
0 0
nd
rd
c. • no.2
C =(2 & 3 part)
2
-1 0
1
1
1
3
0
1
2
-1 0
1
1
• no.3 (2nd part)
d. D =
det(B) =
det(C) =
det(D) =
1
MAT 1092
Mathematical Methods for ICT
3.5 Closure – Lecture 3
1. Importance of determinant of a matrix;
2. How to find out the determinant;
3. Can be used to:
i. Work out inverse of a matrix;
ii. Solve linear equations;
iii. Work out eigenvalues & eigenvectors.
4. Interesting effects on determinant when
Elementary Matrices are multiplied;
5. Tutorial – much expected extra work.
MAT 1092
Mathematical Methods for ICT
3.6 Extra Exercises
10 Question:
0 01
Typical Exam
Let the
0 matrix
1
-2
0A =
0.5
01
and
E1A=
0
2
0
0
8
-4
-8
3
4
1
2
10
1
10
E2A=
0
2
3
2
0
0
5
1
0
1
3
4
0
0
4
E3A=
1
3
4
10
2
0
2
0
0
Determine:
a. E1, E2, E3 (what kind of elementary matrices?)
b. Det(A)
c. Det(E3A)
d. If E2A = A + B find B
e. If F = 3A and det(F) = k.det(A) find k
MAT 1092
Mathematical Methods for ICT
4. Solutions of Linear Equations
1. Introduction
2. Solving linear equations
i. Using Gaussian row reduction
ii. Using inverse of matrix
iii. Using Cramer’s rule
3. Possible solutions
i. Unique
ii. Infinite
iii. None
4. Worked examples
5. Closure
MAT 1092
Mathematical Methods for ICT
4.1 Introduction to Solving Equations
After employing some method to form
equations with the unknown variables,
possible answers have to be figured out.
The determinant and inverse of the matrix are
only used to assist in the solution of the
linear equations.
Possible scenarios:
– Full-valued matrix
– Variables within matrix
MAT 1092
Mathematical Methods for ICT
4.2 Solutions using Gaussian System
Reducing the rows to Echleon form will give
an insight into the values of the unknown
variables.
e.g. Jan 05 no.3 (last part)
3
1 0 0
-2
If A = 52 31 40
solve Ax = 11
R3-5R1
R2-3R3
1 0 0 3
2 3 4 -2
5 1 0 11
1 0 0 3
2 3 4 -2
0 1 0 -4
R2-2R1
1 0 0 3
0 3 4 -8
0 1 0 -4
1 0 0 3
0 0 4 4
0 1 0 -4
R2/4
1 0 0 3
0 0 1 1
0 1 0 -4
 x=3
y = -4
z=1
MAT 1092
Mathematical Methods for ICT
4.3 Solutions using the Inverse Matrix
Employing the inverse of the matrix is an
effective way to solve the linear equations
but slightly more lengthy.
e.g. Jan 05 no.3 (last part)
1 0 0
3
-2
If A = 52 31 40
solve Ax = 11
1 0 0
1. Work out inverse of A ... i.e. A-1 3¼-5 ¼0 -¾
1
2. Apply the1reasoning
0 0multiply
1 0 0
0 0 1 0 0of Ax = 1y &
2 3 4 0 1 0
2 3 4 -10 1 0
both sides5by1 A0 0 0 1
0 1 0 -5 0 1
-1y
 A-1Ax
=
A
1 0 0 1 0 0
1 0 0 1 0 0
0 0 4 13 1 -3
0 -13 4 -2 1 0
 x=A
0 1 0 -5 0 1
0 1y 0 -5 0 1
1 0 0
3
-2
 x = 13¼-50 ¼0 -¾
.
11
0 11 0
3 0 0 1 13/4 ¼ -3/4

x=3
0
1
-4 0 1 0 -5
y = -4
1
z=1
R2-2R1
R3-5R1
R2-3R3
MAT 1092
Mathematical Methods for ICT
R2/4
4.4 Solutions using Cramer’s rule
By far the simplest and straight forward system to
work out the solutions of linear equations is to
employ Cramer’s rule based on the Determinant
of a number of matrices.
1. Separate the two matrices: the linear equation’s
values and answer column;
2. Form additional matrices by inserting the answer
column matrix instead of the unknown variable;
3. Work out the determinants of the new matrices
as well as the original matrix;
4. Divide determinants of each variable matrix with
that of the original matrix.
What if Original Matrix has a determinant = 0
MAT 1092
Mathematical Methods for ICT
4.5 Possible Solutions - Unique
Anyway you decide to solve the linear
equations ... the ideal situation is when a
unique solution is available.
This means that:
i. Gaussian system
–
The identity matrix is obtained on the linear
equation’s side;
ii. Inverse matrix system
–
The inverse matrix exists and is available to
multiply on both sides;
iii. Cramer’s rule
–
The determinant of the main matrix is not equal
to zero.
MAT 1092
Mathematical Methods for ICT
4.6 Possible Solutions - Infinite
The possibility of having an infinite number
of solutions happens when:
i. Gaussian system
–
The identity matrix cannot be achieved on the
main linear equations matrix;
ii. Inverse matrix system
–
The inverse matrix is not possible to find;
iii. Cramer’s rule
–
The determinant of the main matrix is equal to
zero.
MAT 1092
Mathematical Methods for ICT
4.7 Possible Solutions - None
The situation when no solutions are available
for the linear equations occurs when:
i. Gaussian system
–
Inconsistencies are evident in the extended
matrix, like 0 = 1;
ii. Inverse matrix system
–
Even if identity matrix is obtained ... And
inconsistencies are evident even in this case;
iii. Cramer’s rule
–
The determinant of the main matrix is equal to
zero.
MAT 1092
Mathematical Methods for ICT
4.8 Worked Examples
Such problems are very popular examination
problems as they bring out reasoning skills
in students.
Try out from past-papers:
• January 2005 no.1(a)
• January 2006 no.1 (first part)
• February 2007 no.2(b)
MAT 1092
Mathematical Methods for ICT
4.9 Closure - Lecture 4
1. Linear equations can be solved by Matrices
2. Solving linear equations by employing:
i. Gaussian Row Reduction
ii. Inverse of a Matrix
iii. Cramer’s Rule
3. Possible solutions
i. Unique
ii. Infinite
iii. None
4. Worked examples from Past-Papers
5. The undeniable tutorial
MAT 1092
Mathematical Methods for ICT
4.10 Extra Exercises
Typical Exam Question
a 0 b
If A = 0a aa 42
and y =
a 0 b 2
- R1
afind
a 4 the
4 R2values
0 a 2 b
of a and b for which the
system has:
i. A unique solution;
0 bInfinitely
2
If b ≠many
2 then z =solutions;
1 and if a ≠ 0 then x = (2 – b)/a and y = (b – 2)/a
ii.
a 4-b 2
unique solution
0 b-2 b-2
iii. No solutions.
a 0 b 2
0 a 4-b 2
0 a 2 b
a
0
0
2
4
b
R3 – R2
If b = 2 and a = 0 then
0 0 2 2
0 0 2 2
0 0 0 0
Finally if a = 0 and b ≠ 2 then
meaning we have the same lines.
multiple solutions
0 0 b 2
0 0 4-b 2 =
0 0 b-2 b-2
0 0 b 2
0 0 4 4
0 0 1 1
=
0 0 0 1
0 0 4 4
0 0 1 1
no solution
MAT 1092
Mathematical Methods for ICT
5. Eigenvalues & Diagonalisation
1.
2.
3.
4.
5.
6.
Introduction
Finding the Eigenvalue
Working out the Eigenvectors
Determining if a matrix is Diagonalizable
Worked Examples
Closure
MAT 1092
Mathematical Methods for ICT
5.1 Eigen introduction
The Eigenvalue of a matrix is A vector may be
thought of as an arrow.
• It has a length, called its magnitude (scalar value)
• It points in some particular direction.
A linear transformation may be considered to operate
on a vector to change it, usually changing both its
magnitude and its direction.
An Eigenvector of a given linear transformation is a
vector which is multiplied by a constant called the
Eigenvalue during that transformation. The
direction of the eigenvector is either unchanged by
that transformation (for positive eigenvalues) or
reversed (for negative eigenvalues).
Eigenspace corresponding to a given eigenvalue of a
linear transformation is the vector space of all
eigenvectors with that eigenvalue
MAT 1092
Mathematical Methods for ICT
5.2 What are the Eigenvalues?
Given
have a matrix
Find
thewe
eigenvalues
of: A, and this is transformed
when
multiplied by 0vectors
X ...  A.X
2 2
3
1 1
5
1 0
i.When this happens,ii.we2 get0 a-2simple scaling
iii.
1
3 1
1 λ 5 0
-2 2 4
1 3= λ.X
0
0 6
 1 A.X
2, 2,=5(λ.I).X  = 0, 2, 2
 = 4, 4, 6
 =A.X
 = 3, 3, 6
 = 4,=6,0 6
 A.X – (λ.I).X
4
1 1
6
1 0
 (A – λI)
iv. X0 = 06 0
v. 1 4 1
Solutions for 0this0 4are found 1by 1 working
out the
4
determinant of (A-λ.I) and equal to 0
e.g. A = 1 3 so (A-λ.I) = 1 3 -λ. 1 0 = 1- 3
4
2
4
2
0
1
4
det(A-λI) = (1- )(2- ) – (4)(3) = 0
 (1-λ)(2-λ)-12 = 2-3λ+λ2-12 = λ2-3λ–10 = 0
 (λ – 5) (λ + 2) = 0
 λ = 5 and λ = -2 are the Eigenvalues of A
MAT 1092
Mathematical Methods for ICT
2- 
5.3 Find the Eigenvectors
To determine the Eigenvectors of a Matrix, A,
having found out its Eigenvalues is simple.
For every Eigenvalue a respective Eigenvector
will be determined, so recalling from previous
3
slide that if A = 14 32 then (A-λ.I) = 1-
4
2- 
If det(A-λI) = 0 gives Eigenvalues λ = 5 and λ = -2
If the Eigenvector associated with λ = 5 is
Then (A-λ.I) . = 0
1-
3
 4 2-  .
= 0 where λ = 5
x1
x2
x1
x2
x1
x2

-4
4
3
-3
.
x1
x2
=0
 -4x1 + 3x2 = 0 or 4x1 - 3x2 = 0 or 4x1 = 3x2
 If x1 = a then x2 = 4a/3
 Hence Eigenvector
=
or a
MAT 1092
x1
a
1
x2
4a/3
4/3
Mathematical Methods for ICT
5.4 How to determine if a
Matrix is diagonalizable?
A matrix A is diagonalizable if there is an invertible
matrix S such that S-1AS is a diagonal matrix.
It seems like every matrix should be diagonalizable
... All you need to do is:
i.
ii.
•
•
Find the eigenvalues & eigenvectors – construct S by making the
columns of S the eigenvectors of A
Then ensure that S-1AS is possible
But what if S is not invertible?
1
0
...
0
What if its not possible to construct S?

...
0
Also note that if D = S-1AS = 0
So, note that a matrix is diagonalizable
iff:...
0
0

-1 = A
• It has aThen
set ofSDS
linearly
independent eigenvectors
• AIf2 it= has
distinct
eigenvalues
So,
S-1DS
. S-1DS
= S-1D2S
2
n
Also note that:
n = S-1Dn S
Hence,
A
• Every symmetric matrix is diagonalizable
• A matrix is symmetric if it is equal to its transpose
MAT 1092
Mathematical Methods for ICT
5.6 Past-Papers Questions
Eigenvalues & Eigenvectors, due to their
importance, feature in every exam paper,
namely:
• Jan 05
– no.2 (second part)
• Jan 06
– no.1 (last part)
– no.3(b)
Check out the matrices:
A= 2 1
1 2
• Feb 07
– no.1
– no.2(a)(iv)
MAT 1092
B=
7/4
3/4
0
3/4 5/4
0
0
1
Mathematical Methods for ICT
0
5.7 Closure – Lecture 5
1.
2.
3.
4.
5.
6.
Introduced Eigenvalues & Eigenvectors
How to find Eigenvalues
Work out the Eigenvectors
Matrix Diagonalization explained
Worked Examples – past-papers
Typical exam question
MAT 1092
Mathematical Methods for ICT
a) Find Eigenvalues
where
1.5
0
1

0
0
-0.5
0.5
-0.5
0

0
-0.5
0
0
0
0

5.8 Tutorial Stuff
–
Typical
1.5 0- 
0 1
1 x1
When
 = 0.5Exam
then 1Question
=0
Meaning
Det-0.5
of -0.50 and
0.5 -0.5
-
-0.5
x2 = 0
Find
the that
Eigenvalues
Eigenvectors
-0.5
-0.5 0
0 -0.5
 x1 + x3 = 0 meaning that if x1 = a
1.5 0
1
=0
of
-x3
then x3 = -a and x2 = b
a
a
0
0 )) = 0 1

(1.5
)(0.5
)(-)
–
0
+
1(0
–
(-0.5)(0.5
Associated Eigenvector
is
or
+
=a
+b
x2
b
0
0
1
b
 (1.5 - )(0.5
)(-)
–
0
+
1(0
–
(-0.5)(0.5
))
=
0
-0.5 0.5 x-0.5
-a
-1
-a
0
0
3
 (0.5 - )[(1.5 - )(-) + 1( – (-0.5))] = 0
x1 2=
1
When
= 1-then
0 + 0.5) = 0
-0.5 0.5+02 + 00.5)
0 = (0.5
(0.5
)(-1.5
- )(
-1.5
-0.5
 (0.5 - )( - 1)(
- 0.5)-0.5
= 0 -0.5 x2
x3
-0.5
0
-1
itdiagonalizable?
= 0.5 OR  = 1 OR  = 0.5 → eigenvalues
Is
 0.5x1 + x3 = 0 and x1 + x2 + x3 = 0 meaning that if x1 = a
then x3 = -0.5a and x2 = -0.5a
b) Find Eigenvectors for these Eigenvalues
Associated Eigenvector x1 is a
or a 1
Meaning that for each Eigenvalue
we
x2
-0.5aapply:
-0.5
1.5 - 
x1
x3
0
-0.5a
1
.
-0.5S-1 and
0.5D.
-
-0.5
c) Now we can find S,
1 0 1
-0.5
MAT 1092
0
x1
x2
0 - 1 -½x3
-1 0 -½
-0.5
=0
-1 0 -2
1 1 1
2 0 2
Mathematical Methods for ICT
½ 0 0
0 ½ 0
0 0 1
6. Ordinary Diff. Eqns. Of 1st. Order
1.
2.
3.
4.
Introduction
Differentiation & Integration
Points of reference – general rules
First order differential equations
a. Separable
b. Exact
c. Integrating Factor
5. Exam Questions
6. Closure
7. Practice examples.
MAT 1092
Mathematical Methods for ICT
6.1 Introduction
Two type of differential equations will be
covered:
1. First-order
– Simple ones
– Slightly more complicated one
2. Second-order (Lecture 7)
Still ... the basic concepts have to be known
and understood right away.
MAT 1092
Mathematical Methods for ICT
6.2 Differentiation & Integration
They are the inverse of each other...with a slight twist.
Differentiation of a function gives the measure of a
slope thereby finding its derivatives.
Integration reverses this process and gives the function
once you know the derivatives.
E.g. y = f(x) = 3x2 + 4x + 2
Then its differentiation is dy/dx = y’ = f ’(x) = 6x + 4
If we had to integrate 6x + 4 we should get y back.

 y’ dx =  dy/dx dx = 1dy = y

(6x + 4) dx = 6x2/2 + 4x/1 + c (constant)

3x2 + 4x + c
This is simple and easy ... Slightly more complicated
derivatives and their corresponding integrals follow.
MAT 1092
Mathematical Methods for ICT
6.3 Differential Rules
3). What
•e.g.Constant
If f(x)
=
k,
then
f’(x)
=
0.
supposeRule:
y = ln(x
is
dy/dx?
3x2, what
e.g. Suppose f(x) = e2x
is What
f’(x)?is f’(x)?
/
(x-2).
e.g. Suppose f(x) = 3. What is f’(x)?
Try out
n, then f’(x) = cnxn-1
2
•Try
Power
Rule.
If
f(x)
=
cx
out
out:notorious
a)25x derivatives:
/ 5x
Some
a) Suppose
Whatisisf’(x)?
f’(x)?
e.g.
Supposef(x)
f(x)==ln(x).
3x2, what
If y =b)7e6 /what
is dy/dx?
x Rule:
• a)
Sum-Difference
If
f(x)
=
g(x)
±
h(x)
2
b)
Suppose
f(x)
=
log(3x
+
x
).
What
is
f’(x)?
Trythen
out f’(x)
the following:
2 3+2lnx
3 –e
g’(x)
h’(x)
b) If f(x)c)==(16x
5x
+±12x)
/22x
4 ln(x3)
ln
xRule:
then
= dy/dx = 1/x
c)y =Suppose
y =If
3x.ln(5x
+y’2x)
• a)IfProduct
f(x)
=
g(x).h(x)
3
10x out:
Try
f’(x)=g’(x).h(x)+g(x).h’(x)
Ifthen
y = sin
x
then
y’ = dy/dx = cos x
• b) Quotient
If f(x)=g(x)/h(x)
a)6xf(x) = Rule:
17 – 4x
2
3
2
then
f’(x)=[g’(x).h(x)-g(x).h’(x)]
/
[h(x)]
If
y
=
cos
x
then
y’
=
dy/dx
=
-sin
x
e.g.
Suppose
f(x)
=
(4x
)(5-x
).
What
is
f’(x)?
2
c) b)2xy += 112x3 – 4x2 + 5x2 – 7
e.g.
Suppose f(x) = 2x / (x-2). What is f’(x)?
Try
out:
2x
If
y
=
tan
x
then
y’
=
dy/dx
=
sec
• Log
If 6x
f(x) = ln( g(x) )
c) yRule:
= 3x2 – 4 +
a) y f’(x)
= 6x(2x
– 5) / g(x)
then
=
g’(x)
x
If y = ex
then
y’
=
dy/dx
=
e
b) f(x) = (12x3 – 4x2 +Rule:
5x – If
7)(3x
• Exponential-Function
f(x) –=7)
eg(x)
g(x)
then
f’(x)
= g’(x).e
c) f(x)
= (17
– 4x)(5)
MAT 1092
Mathematical Methods for ICT
6.4 Separable FODEs
Simplest of all where the differential equation
can independently be separated.
function of x  function of y
 A(x) dx + B(y) dy = 0
 A(x).dx + B(y).dy = c
e.g. Solve the following FODEs
i. x.dx – y2.dy = 0
ii. y’ = y2x3
iii. dy/dx = (x2 + 2) / y
MAT 1092
What if y = 2 when x = 0
Mathematical Methods for ICT
6.5 Exact FODEs
If a FODE is not separable then it needs to be sorted in
some other way.
Consider 2xy.dx + (1 + x2).dy = 0
In this case we have two functions both of which are
of x and y ... A(x,y) & B(x,y)
All we need to do is to find a function to group them
together  g(x,y) such that:
dg(x,y) = A(x,y).dx + B(x,y).dy
i. Check if function is exact  dA/dy = dB/dx
If function is not exact ... what happens?
ii.
Choose
to
work
out
dA/dy
or
dB/dx
ii. Either use y=xt or multiply by Integrating Factor
iii. Find gasbybefore
integration and equate to other derivative
iii.Solve
e.g. Determine if exact and solve the following FODE
i. 2xy.dx + (1 + x2).dy = 0
ii. (x + sin y).dx + (x.cos y – 2y).dy = 0
iii. y’= (2y4 + x4) / xy3
MAT 1092
Mathematical Methods for ICT
6.6 Integrating Factor
When the equation is not exact or separable then we
multiply it throughout with a factor to simplify it
 Integrating Factor.
Consider the equation A(x,y).dx + B(x,y).dy = 0
The Integrating factor, I = e J(x).dx or I = e J(y).dy iff:
J(x) = (dA/dy - dB/dx) / B or J(y) = (dB/dx - dA/dy) / A
Multiplying all the terms by I will produce an exact
differential equation.
 I.A(x,y).dx + I.B(x,y).dy = 0
e.g. Consider y’ + (3xy + y2)/(x2 + xy) = 0
 (3xy + y2).dx + (x2 + xy).dy = 0
dB/ = 2x + y
 dA/dy = 3x + 2y
and
dx
 (dA/dy-dB/dx)/B=1/x & (dB/dx-dA/dy)/A=-(x+y)/(3xy+y2)
 Cont ...
MAT 1092 Mathematical Methods for ICT
6.7 Typical Questions
• Jan 05
Section B
Sample
questions:
no.1
1. –Solve:
–a) no.3(b)
y’ = y2x3
= (x2 + 2)/y
• b)
Jandy/dx
06 Section
B
y’ = (a)
5y & (b)
–c) no.4
y’ = (x + 1)/(y4 + 1)
–d) no.6(b)
2 + 9).dt
e)
dy
=
2t(y
• Feb 07 Section B
f)
g)
h)
i)
j)
y’ = (y + x)/x
y’ = 2xy/(x2 - y2)
y’ = (x2 - y2)/xy
y’=y2/[xy+(xy2)1/3]
y’ = y/ x2
– no.1(a)
2. Find the Integrating factor for:
– no.2(a)
a) y’ + 4y/x = x4
– no.3(c)
b) y’ – 2xy = x
c) z’ - 2z/x = 2x4/3
3. Solve the FODEs in 2.
MAT 1092
Mathematical Methods for ICT
6.8 Closure – Lecture 6
1.
2.
3.
4.
5.
Introduced Differentiation & Integration
Rules and important derivatives/integrals
Solved separable differentials
Identified exact differential forms
Explained and worked out the Integrating
factor
6. Typical exam questions
MAT 1092
Mathematical Methods for ICT
6.9 Practice Stuff
Typical Exam Question
a) Solve the differential equation
y’ = x2 + y2
xy
b) Show that the differential form is not exact
y’ + 6xy = 0
c) Find the integrating factor for the equation
in b) and work out the integral
MAT 1092
Mathematical Methods for ICT
7. Ordinary Diff. Eqns. Of 2nd. Order
with Constant Coefficients
1. Introduction – characteristic equation
2. Solution to SODEs – 3 cases
– First case
– Second case
– Third case
3. Typical Questions
4. Closure – Lecture 7
5. Practice examples
MAT 1092
Mathematical Methods for ICT
7.1 Introduction to SODEs
A Second Order Differential Equation (SODE) is an
equation that has the application of
Differentiation twice within it.
e.g. If y = 5x4 + 2x3 - 7x2 + 4x – 12
D = dy/dx = y’ = 20x3 + 6x2 - 14x + 4 (FODE)
2
2=d
(y’)
d
dy
dy
D /dx = /dx( /dx)= /dx2= y”= 60x2+12x–14 (SODE)
So a SODE will be of the form: y’’+3y’ - 4y = 0
Thereby we replace the derivative of y by powers of
the differential operator, D.

Characteristic Equation: D2 + 3D - 4 = 0
This equation can then be factorised ...
 D2 + 3D - 4 = (D + 4) (D - 1) = 0
 D = -4 OR D =1 meaning y’ = -4 OR y’ = 1
MAT 1092
Mathematical Methods for ICT
7.2 Solution to SODE
The solutions of differential equations
involving Second Order Differential
Equations are not straight forward and
depends entirely on the roots of the
factorised Characteristic Equation.
Case 1: If the roots are real;
Case 2: If the roots are complex;
Case 3: If the roots are real and equal to each
other.
MAT 1092
Mathematical Methods for ICT
7.3 Solving SODEs – First Case
If the roots of the Characteristic Equation are
real and distinct, then the solutions are eD1x
and eD2x where D1 and D2 are the roots.
e.g. y’’+3y’ - 4y = 0
 D2 + 3D - 4 = 0
 (D + 4) (D – 1) = 0
 D = -4 or D = 1
 Roots are real and distinct
 Solutions for y = c1e-4x + c2e1x
 y = c1e-4x + c2ex
MAT 1092
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7.4 Solving SODEs – Second Case
If the roots of the Characteristic Equation are
complex, then the solutions are e(a+ib)x and
e(a-ib)x where (a+ib) and (a-ib) are the roots.
e.g. y’’+ 4y = 0
 D2 + 4 = 0
 (D + 2i) (D – 2i) = 0
 D = -2i or D = 2i
 Roots are complex with a = 0 and b = 2
 Solutions for y = c1e-2ix + c2e2ix
 OR also y = c1e0xcos2x+ c2e0xsin2x
 y = c1cos2x+ c2sin2x
MAT 1092
Mathematical Methods for ICT
7.5 Solving SODEs – Third Case
If the roots of the Characteristic Equation are
real and equal, then the solutions are eD1x
and xeD1x where D1 is the root.
e.g. y’’- 8y’ + 16y = 0
 D2 – 8D + 16 = 0
 (D - 4) (D – 4) = 0
 D = 4 or D = 4
 Roots are real and equal
 Solutions for y = c1e4x + c2xe4x
MAT 1092
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7.6 Typical SODE Questions
Solve
1. y” – y’ – 2y = 0
2. y” – 7y’ = 0
3. y” – 5y = 0
4. y” – 6y’ + 25y = 0
5. d2y/dx2 + 4dy/dx + 5y = 0
MAT 1092
Mathematical Methods for ICT
7.7 Closure – Lecture 7
1. SODE are equations involving not only
derivatives but also derivatives of
derivatives.
2. Solution to SODEs – 3 cases
– Real & Distinct roots
– Complex roots
– Equal roots
MAT 1092
Mathematical Methods for ICT
7.8 Practical SODE Examples
Let D denote the operator of differentiation,
compute the following SODE:
1. y” – y = 0
2. d2x/dt2 – 10dx/dt + 25x = 0
3. y” + y’ + 1/4 y = 0
MAT 1092
Mathematical Methods for ICT
8. Partial Differential Equations and
Exact Differential Equations
1.
2.
3.
4.
5.
6.
7.
8.
Introduction – undetermined coefficients
Case 1 – nth degree polynomial
Case 2 – ex
Case 3 – sin and cos
Definition of Partial DE
Definition of Exact DE
Closure
Final problems to solve
MAT 1092
Mathematical Methods for ICT
8.1 Introduction
The solutions to the SODE given in the
previous section all equated to zero.
What if the SODE equated to something
different from zero, like:
Case 1: SODE = f(x)
Case 2: SODE = kenx
Case 3: SODE = k1sin nx + k2cos nx
where k and n are all constants
MAT 1092
Mathematical Methods for ICT
8.2 Special SODEs – First Case
If the SODE = f(x) then a solution is of the form
Anxn + An-1xn-1 + An-2xn-2 + ... + A1x + A0
where A is a constant
If y”-y’-2y = 0
e.g. y”- y’ - 2y = 4x2
 Solution: y = A2x2 + A1x + A0 y = c e-x + c e2x
1
2
 y’ = 2A2x + A1 and y” = 2A2
Substituting into original equation:
 2A2 – (2A2x + A1) – 2(A2x2 +A1x +A0) = 4x2
 (-2A2)x2+(-2A2-2A1)x+(2A2–A1–2A0)=4x2+(0)x+0
Equating coefficients of powers of x:
 -2A2 = 4 and -2A2-2A1 = 0 and 2A2–A1–2A0 = 0
 A2 = -2 and A1 = 2 and A0 = -3
 y = c1e-x + c2e2x - 2x2 + 2x - 3
MAT 1092
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8.3 Special SODEs – Second Case
If the SODE = kenx then a solution is of the form
Aenx
where A and n are constants
e.g. y”- y’ - 2y = e3x
If y”-y’-2y = 0
 Solution: y = Ae3x
 y’ = 3Ae3x and y” = 9Ae3x
y = c1e-x + c2e2x
Substituting into original equation:
 9Ae3x - 3Ae3x - 2Ae3x = e3x
 4Ae3x = e3x
 4A = 1 or A = ¼
 y = c1e-x + c2e2x + ¼ e3x
MAT 1092
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8.4 Special SODEs – Third Case
If the SODE = k1sin nx + k2cos nx then a solution is
of the form
A sin nx + B cos nx
where A and n are constants
e.g. y”- y’ - 2y = sin 2x
If y”-y’-2y = 0
 Solution: y = Asin2x + Bcos2x
y = c1e-x + c2e2x
 y’ = 2Acos2x – 2Bsin2x
 y” = -4Asin2x – 4Bcos2x
Substituting into original equation:
 (-4Asin2x-4Bcos2x)-(2Acos2x-2Bsin2x)-2(Asin2x+Bcos2x)=sin2x
 (-6A+2B)sin2x + (-6B-2A)cos2x = (1)sin2x + (0)cos2x
Equating coefficients of like terms:
 -6A + 2B = 1
and
-2A – 6B = 0
 A = -3/20
and
B = 1/20
 y = c1e-x + c2e2x – 3/20sin2x + 1/20cos2x
MAT 1092
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8.5 Final DE definitions
A differential equation is an ordinary differential
equation if the unknown function depends on
ONLY 1 independent variable.
If the unknown function depends on 2 or MORE
independent variables, the differential equation is a
partial differential equation.
Finally, those differential equations which out of their
nature are solved without any manipulation are
called exact differential equations.
MAT 1092
Mathematical Methods for ICT
8.6 Closure – Lecture 8
1. SODE that do not equate to zero have been
investigated and solution has been given in
the case it equates to:
– A function of the unknown variable
– An exponential of the unknown variable
– A combination of Sin & Cos of the
unknown variable
2. Final definitions of DE were given, namely:
– Ordinary
– Partial
– Exact
MAT 1092
Mathematical Methods for ICT
8.7 Practical
Solve:
1. y” – 6y’ + 25y = 2sin(t/2) – cos(t/2)
2. d2y/dx2 – 4dy/dx + 4y = 4e2x
3. y” – 2y’ + y = x3 + 2x
4. d2y/dx2 + 16y = sin 4x
MAT 1092
Mathematical Methods for ICT
9. Closure
1.
2.
3.
4.
5.
6.
Introduction
Matrices
Determinants
Solution of linear equations
Eigenvalues and diagonalisation
Ordinary Differential equations of the first
order
7. Ordinary differential equations of the
second order with constant coefficients
8. Partial differential and exact differential
equations.
9. Closure
MAT 1092
Mathematical Methods for ICT
Past Papers
1.
2.
3.
4.
5.
6.
Introduction
Matrices
Determinants
Solution of linear equations
Eigenvalues and diagonalisation
Ordinary Differential equations of the first
order
7. Ordinary differential equations of the
second order with constant coefficients
8. Partial differential and exact differential
equations.
9. Closure
MAT 1092
Mathematical Methods for ICT