Transform of direvativas
Theorem.
Example 1.
Then solution of initial-value problem is
Example 2.
Hence a solution of given equation
Theorem.
Example 1.
Theorem.
Example 1.

Eigenvalues and eigenvectors.
Definition.
Let
eigenvalue of
A
A= [ aij ]
be an n n matrix. The number
is called
n
if there exists a nonzero vector x R such that
Ax= x ( 1 ) .
Every nonzero vector x
satisfying
associated with the eigenvalue
A
(1) is calle

Diagonalization
Definition.
B
A matrix
P
nonsingular matrix
is said similar to a matrix
A
if there is a
such that
1
B=P A P .
[
A= 1 1
2 4
Example 1. Let
A 11 =2
A 12=1
,
[
and
,
A 21=1
]
and
P1= 2 1
1 1
[
]
][
[ ] . Then
P= 1 1
1 2
|P|=1,
, A 22=1
][ ] [

Transition matrices
Suppose now that
n
v
w
T =( 1 , w2 , , w n)
S=( 1 , v 2 , , v n) ,
dimensional vector space V . We shall examine the relationship between
the coordinate vectors
S
are bases for the
[ u ] S and [ u ] T of the vector u
in
V
to the bases

The rank of a Matrix and Application
[
a11 a12 . a1 n
a21 a 22 a2 n
A= .
ai 1 a i2 a
.
am 1 am 2 amn
Definition. Let
The rows of
]
be an m n matrix.
A,
a
( 11, a12, , a1 n )
v 1=
a
21
,
a
(
22, , a2 n)
v 2=
.
a
m1
,
a
(
m 2, ,a mn)
v m=
considered as vec

Properties of Matrix operations.
Theorem 1. Properties of Matrix Addition
For m n matrices A, B, and C, the following properties hold.
Closure property: A + B is again an m n matrix.
Associative property: (A + B) + C = A + (B + C).
Commutative property: A

The inverse of a matrix
Definition.
A= [ aij ]
Let
( n1 ) (n1)
A
sub matrix of
A
th column of
M ij
be an n n matrix. Let
obtained by deleting the
.
[
]
a11 a12 a1 j a 1n
a 12 a22 a 2 j a2 n
M ij = . ith row
ai 1 ai 2 aij a
.
an 1 a n 2 a n j an n
j
The d

Vector spaces
V
Definition 1. A real vector space is a set of elements
together with two
operations satisfying the following properties:
.
u
If
and v are any elements of
a). u v=v u
u
for any
u ( v w ) =( v u) w
b).
c). There is an element 0
u
d). For ea

Basis and dimension vector spaces.
Definition 1. If
S=cfw_v 1 , v 2 , , v k
is a set of vectors in a vector space V ,
V
then the set of all vectors in vector space
vectors in
S
is denoted by the span S or
that are linear combination of the
span cfw_v 1 ,

Linear Systems.
Many problems in natural and social sciences as well as in engineering and
physical sciences deal with linear system of equations.
An equation of the type
ax=b
b
expressing the constant
in terms of the variable x
and constant a is
called
a

Matrices
m n
Definition. A
complex) numbers arranged in m
[
A
size matrix
is a rectangular array of m n real (or
n
horizontal rows and
vertical columns
]
a11 a12 a1 j a1 n
a 12 a22 a2 j a 2 n
A= . ithrow ( 1 )
ai 1 ai 2 aij a
.
am 1 am 2 amj amn
j th co

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Name of Course / Mode
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The matrix of linear transformation
Theorem 1.
space
V
Let
L:V W
be a linear transformation of n dimentional
into an m dimentional space
S=cfw_v 1 , v 2 , , v n T =cfw_w1 , w2 , , wm
Then the m n matrix
[ L(v j)]T of
L(v j)
A
W
(n 0m 0)
and let
be bases

Exact equations
z=f ( x , y )
If
is a function
of two variables with continuous
first partial
derivatives in a region R of the xy plane, then its differential is
d z=
If
For example, if
f
f
dx +
dy (1)
x
y
f ( x , y ) =c then dz=0(2)
x 25 xy+ y 3=c , then

Differential equations
Definition 1. (Differential equation)
An equation containing the derivatives of one or more depended variables,
with respect to one or more independent variables, is said to be a differential
equation.
Definition 2. If an equation c

LAPLACE TRANSFORMATION.
In elementary calculus you learned that differentiation and integration are
transforms; this means, that these operations transform a function into another
function. For example, the function is transformed, in turn, into a linear

Power series, solution about ordinary points.
If L<1 then the power series converges at x. Radius of convergence: R=lim|cn+1/cn| as n tents to
infinity.
Suppouse the linear second order differential equation
'
'
a2 (x ) y + a1 ( x) y + a0 (x) y=0 (5)
y '

Solution about singular points.
Definition.
Theorem.
Example 1. Use Frobenius method to solve given differential equation about a regular
singular point
'
'
x=0, 3 x y + y y =0 .
Solution. We will try to find a solution of the form
Then
So
Then
And
y=C 1

MODELING WITH HIGHER-ORDER
DIFFERENTIAL EQUATIONS.
1. SPRING/MASS SYSTEMS:
FREE UNDAMPED MOTION
HOOKES LAW Suppose that a flexible spring is suspended vertically from a
rigid support and then a mass m is attached to its free end. The amount of stretch,
or

Cauchy-Euler equation
Definition .
A linear differential equation of the form
an x n y (n) +an1 x n1 y (n1) + a1 x y ' + a0 y=g ( x )
an , an1 , , a0
where the coefficients
are constants, is called a
Cauchy-Euler
equation.
We start the discussion with a d

Variation of parameters
Let given a linear second-order differential equation
'
'
a2 (x ) y + a1 ( x) y + a0 (x) y=g (x)(1)
For the linear second-order differential equation
solution in the form
(1) we seek a particular
y p=u1 ( x ) y 1 +u2 ( x ) y 2
wher

Reduction of order
1. In case equation on form
(n)
=f (x)
2. In case the dependent variables
that is
solved by n times integrating.
is missing from our differential equation,
x , ' , ) =0
, we make the substitution
F
'=
. This entails
= p ' .
The diffe

a
Homogeneous linear equations with constant coefficients
Definition. DE
(n )
an y + an1 y
(n1)
'
'
+a 2 y + a1 y +a 0 y=0(1)
is said to be norder homogeneous linear differential equation with constant
coefficients if the coefficients
ai , i=1,2, ,n
are r

Linear equations
Definition 1.
(linear equation)
A first-order differential equation of the form
a1 ( x )
dy
+a ( x ) y=g ( x ) (1)
dx 0
is said to be a linear equation in the dependent variable
When
g ( x ) =0
y
.
, the linear equation (1) is said to be

Linear transformations
Definition.
Let
V
and W be vector spaces.
L
A linear transformation
L(u)
vector
in
W
to each u
a)
L ( u+ v )=L ( u ) + L(v)
b)
L ( ku )=kL ( u )
V =W
If
of
V
in
V
u V
, the linear transformation
is a function assigning a unique
such

Homogeneous equations
If a function f
Definition 1.
possesses the property
f ( tx , ty ) =t f ( x , y)
for some real number , the f
is said to
be homogeneous function of degree .
For example,
f ( x , y ) =x3 + y 3
is homogeneous function of degree 3 ,
f (

INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA
MIDTERM EXAMINATION
SEMESTER II, 2012/2013 SESSION
KULLIYYAH OF ENGINEERING
Name
:
Q1
Matric No
:
Q2
Section
:
Q3
Date
: 23/03/2013
Q4
Duration
: 2 Hrs
Time
: 05-07 pm
Q5
Course Code : MTH 2132\2311
Section(s):

Transform of direvativas
Theorem.
Example 1.
Then solution of initial-value problem is
Example 2.
Hence a solution of given equation
Theorem.
Example 1.
Theorem.
Example 1.

halls. MHHELAQLiiii-éeam“
Q9 .9 9.
INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA
END OF SEMESTER EXAMINATION
SEMESTER I, 2011/2012 SESSION
KULLIYYAH OF ENGINEERING
Programme : ENGINEERING Level of Study : UG 2
Time : 2:30 pm - 5:30 pm Date :03 / 01 /2012
Dura

Diagonalization
Definition. A matrix B is said similar to a matrix
nonsingular matrix P such that
if there is a
B = P 1 A P.
1
Example 1. Let = [
2
11 = 2,
12 = 1 ,
2
1
2
B = P 1 A P = [
1
2 0
[
].
0 3
P 1 = [
1
] and
4
1
=[
1
1
]. Then | = 1,
2
21 = 1 ,