Review of Power Series
Real Analytic functions
Ordinary points of a 2nd Order
Homogeneous L.D.E.
In this lecture we discuss the Power series
method of solving a second order
homogeneous linear differential equation
First we spend some time in reviewing th
Fourier Series
24.11.12
2005-2006, Nadeem-ur-Rehman
We will see that many important problems involving
partial differential equations can be solved,
provided a given function can be expressed as an
infinite sum of sines and cosines.
In this section and in
Inverse Laplace Transform:
If L[f(x)] = F(p), then f(x) is called an inverse
Laplace transform of F(p), and we write
f(x) = L-1[F(p)].
1
1
1[ ] = 1.
L[1] = = L
p
p
n ] = n! = L1[ n! ] = x n
L[ x
n+1
n+1
p
p
1
xn
L1[
]=
n!
p n+1
24.11.12
2005-2006, Nadee
Laplace Transform
24.11.12
2005-2006, Nadeem-ur-Rehman 1
In this chapter we will introduce a technique
as the Laplace transform, which is a very
useful tool in the study of linear differential
equations. Although be no means limited to
this class of probl
HOMOGENEOUS LINEAR
SYSTEM WITH CONSTANT
COEFFICENTS:
TWO EQUATIONS IN TWO
UNKNOWN FUNCTIONS
11/24/12
2005-2006, MATH C241 Prepared by Nadeem-ur-Rehma
1
We Shall concerned with the homogeneous
linear system
dx
= a1 x + b1 y
dt
dy
= a2 x + b2 y
dt
(1)
whe
Introduction to Systems of First
Order Linear Equations
2005-2006, Nadeem-ur-Rehman 1
A system of simultaneous first
order ordinary differential
equations has the general form
x1 = F1 (t , x1, x2 ,K xn )
x2 = F2 (t , x1, x2 ,K xn )
M
xn = Fn (t , x1, x2
Review of Power Series
Real Analytic functions
Ordinary points of a 2nd Order
Homogeneous L.D.E.
In this lecture we discuss the Power series
method of solving a second order
homogeneous linear differential equation
First we spend some time in reviewing th
Q ualitativePropertiesofsolutions
ofasecondorderhomogeneous
LinearDifferentialequations.
Throughout this chapter we shall be
looking at the second order homogeneous
linear differential equation
y + P( x) y + Q( x) y = 0 .(1)
We shall like to say something
We assume a particular solution of
Ly = k1e cos bx + k 2e sin bx
ax
ax
as y = y p = A1e cos bx + A2e sin bx
ax
ax
We assume a particular solution of
ax
n
Ly = e (b0 + b1 x + . + bn x )
as y = y p = e ( A0 + A1x + . + An x )
We also use multiply by the lea
Particular Solutions of Non-Homogeneous
Linear Differential Equations with constant
coefficients
Method of Undetermined
Coefficients
In this lecture we discuss the Method of
undetermined Coefficients.
11/24/12
1
Consider the second order non-homogeneous
l
Second Order Linear Differential equations
In this lecture
We define a second order Linear DE
State the existence and uniqueness of solutions
of second order Initial Value problems
We find the general solution of a second order
Homogeneous Linear DE
D
LinearDifferentialEquations
LinearDifferentialEquations
In this lecture we discuss the methods
of solving first order linear differential
equations and reduction of order.
LinearEquations
LinearEquations
Alinearfirstorderequationisanequation
thatcanbeexpr
FIRST ORDER
DIFFERENTIAL EQUATIONS
In this lecture we discuss various methods
of solving first order differential equations.
These include:
Variables separable
Homogeneous equations
Exact equations
Equations that can be made exact by
multiplying by an
DifferentialEquations
DifferentialEquations
MATHC241
Class hours: T Th S 2
(9.00 A.M. to 9.50 A.M.)
TextBook:DifferentialEquations
withApplicationsand
HistoricalNotes:
by George F. Simmons
(Tata McGraw-Hill) (2003)
In this introductory lecture, we
Define