3 - laplace transform
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CHAPTER 3 :
LAPLACE TRANSFORMS
3.0 Introduction
3.1 Table of Laplace Transform
3.2 The Inverse of Laplace Transform
3.3 Solving Differential Equations by
Laplace Transform
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3.0 INTRODUCTIONIn this chapter, we will be looking at how to use Laplace
transforms to solve differential equations.• An equation in which at least one term contains etc.
is called differential equation . For example; – → first order differential equation ;
– → second order differential equation
There are many kinds of transforms out there in the world.Laplace transforms & Fourier transforms are probably the maintwo kinds of transforms that are used. As we will see in later sections we can use Laplace transforms toreduce a differential equation to an algebra problem.The algebra can be messy on occasion, but it will be simpler thanactually solving the differential equation directly in many cases.The Laplace transform is commonly used in theprocess of modeling and control.It is often preferred for the solution of
differential equations in engineering & electronics
ydx
dy x 43
2
2
2
43 x y
dx
dy
dx
yd
2
2
,dx
yd
dx
dy
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3.0 INTRODUCTION_cont.
In this topic, we discuss the meaning of the transform and itsoperations that can be applied to solve differential equations.
Laplace transform provide an algebraic method of obtaininga particular solution of a differential equation from stated initialcondition.
The Laplace transform of a function f (x ), denoted byis defined as the function F(s) by the equation;
x f L
where s is a variable whose values are chosen so as to ensure that thesemi-infinite integrals converges.
0
dx x f esF x f L sx
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Example 1: .0,2 x x f Find the Laplace transform of the function
ss
s
edx e
dx eL
dx x f e x f L
sx sx
sx
sx
21
02
22
22
00
0
0
x e sx as
Solution
Step 1 Use the definition:
Step 2 Substitute with thef(x)
:
Notice that s > 0 is demanded because if s < 0 then
& if s = 0, then L{2} is not defined, so that; .0,2
2 ss
L
0, ss
a
aLBy the same reasoning, if a is some constant then;
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Example 2:
asas
as
edx e
dx eeeL
x as
x as
ax sx ax
110
00
0
0, x e x f ax Find the Laplace transform of the function
where a is a constant
Solution
Step 1 Used the definition:
Step 2
Notice that s + a > 0 is demanded to ensure that the integralconverges at both limits, so;
asasas
eL ax
providedisthat,0,1
Simplify & Integrate:
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3.0 INTRODUCTION_cont.
For the Laplace transform to exist the integrandmust converge to 0 as x ∞ and this will impose some
conditions on the values of s for which the integral doesconverge & hence the Laplace transform exists.
The transform produces several changes in the equation:
The last point is the biggest single reason the Laplacetransform is valued
It transforms linear differential equationsinto algebraic equations, which many peoplefind easier to solve.
x f e sx
Variable is t (time) Variable is s (dimensions of inverse time)
t is Real number s is Complex number
Solutions from "Time Domain“ Solutions from "Laplace Domain"
Differential equation Algebraic equation
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Transform
Inverse
Transform
Inverse
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Practice 1:Determine the Laplace transform of each of the following:
a) f(x) = -3b) f(x) = e2x
c) f(x) = -5e-3x
d) f(x) = 2e7x-2
e) f(x) = sin 3x
Solution
s
L x f L33
a)
Use table,F1
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3.2 THE INVERSE OF LAPLACE
TRANSFORM
sF L x f 1
The Laplace transform is an expression in the variable s whichis denoted by F(s) . It is said that f(x) and F(s) = L{f(x)} form a transform pair.
This means that if F(s) is the Laplace transform of f(x) then
f(x) is the inverse Laplace Transform of F(s) . We write;
There is no simple integral definition of the inverse transform
so we have to find it by working backwards. For example;
s
sF x f L x f 4
4
44 1 x f sF L
s
sF
The Laplace
transform
The Inverse
Laplace transform
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Example 3: ?
1
1
s
sF What is the inverse Laplace transform of
Solution
Step 1 Refer to the table, read from right to left, fit the form in table:
Inverse
Step 2 Adjust the function based on the given transform, & identifythe inverse:
x
e
sLsF L x f
ssF
1
1
1
1
11
a = 1
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Practice 2:Determine the inverse Laplace transform of each of the following:
a) F(s) = 3/sb) F(s) =
c) F(s) =
d) F(s) =
e) F(s) =
23
s
22
5
s
s
5
1
s
4
122 s
Solution
3
311
x f
sLsF La)
Use table, F1 (read from
Right to Left)
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3.3 SOLVING DIFFERENTIAL
EQUATIONS BY LAPLACE
TRANSFORMThe solutions of differential equations are the particularsolution of the equations subject to the given
conditions.We need to translate a differential equation into analgebraic form, which can in turn be translated into thesolution of the differential equation. Thus, we can solve adifferential equation by using algebra and specificalgebraic forms.
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0
0
f x f sL
f ssF x f L
00
00
2
2
f sf x f Ls
f sf sF s x f L
Firstderivative
The Laplace transform of the first and secondderivative of a function are given as:
3.3.1 The Properties of Laplace
Transform
Secondderivative
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Example 4: x f x f 21. Given that , express in
terms of s and the transform of f ( x ). 1000 f and f
Solution
Use the linear property & the transform of derivative;
12
21
0210
020022
2
2
2
2
f Lss
f sLf Ls
f sLsf Ls
f f sLf sf f Lsf Lf L x f x f L
First
derivative
Second
derivative
Substitute withthe given value
Factorize
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Example 4:2. If , determine f ( x ). 0122 f Lss
ss
f L
f L )ss(
2
1
0122
2
x
e
ssL
ssL
ssL x f
2
1
1
2
1
12
1
2
2
2
1
21
2
1
Solution
Rearrange the equation;
Make L(f) as asubject
Fit the form intable of LP : F7
Step 1
Step 2 To determine f(x) , we need to solve for the inverse.
a = 2
ass
a
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Example 4:3. Solve .00where2 f x f x f
12
21
220
2
ssf L
sf Ls
sf Lf sL
Lf Lf f sL
L x f L x f L
Solution
Taking Laplace Transform on both sides of the equation.
First
derivative
Step 1
Transform fromthe table of LP
: F1
Make L(f) as a subject
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Example 4:3. Solve .00where2 f x f x f
11
2
s
B
s
A
ss
1222
1
12
2
1
22
1
22
11
1
x x ee
sL
sL
ssL x f
ssf L
Solution
To find the inverse, we need to fit forms in the table:Express as
Partial Fraction
Fit the form intable of LP :
F1 & F4
Step 2
Step 3 Substitute & determinethe inverse transform.
22
1212
;1;012
B A
B A
ssBss AThen,
If
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Example 5:1. Solve the differential equation .10i02 f f x f x f
12
2
212
0122
002
02
sf L
f Ls
f Lf sL
f Lf f sL
L x f L x f L
Solution
Taking Laplace Transform on both sides of the equation.
First
derivative
Step 1
Make L(f) as a subject
x e
s
L
sL x f
sf L
21
21
1
1
1
2
2
12
2
122
To find the inverse, we need tofit forms in the table:
Fit the form intable of LP :
F4
Step 2
Factorize 2
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Example 5:2. Using Laplace transform, solve the differential equation
.20and10if cos
y y x y y
1
2
11
21
1
12
121
cos00
cos
2222
2
2
2
2
2
2
2
ss
s
s
sf L
ss
sf Ls
s
sy Lsy Ls
s
sf Lsy Ls
x Ly Ly sy y Ls
x L x y L x y L
Solution
Taking Laplace Transform on both sides of the equation.
Secondderivative
Step 1
Transform fromthe table of LP
: F10
Make L(f) as a subject
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Example 5:
x x x x
sLs
s
Ls
s
L
sL
s
sL
s
sL x f
ss
s
s
sy L
sin2cossin2
1
1
1
211
2
2
1
1
12
11
1
2
11
2
1
2
1
22
1
2
1
2
1
22
1
222
2
Solution
To find the inverse, we need to fit forms in the table:
Fit the form intable of LP : F18,
F10 & F11
Step 2
2. Using Laplace transform, solve the differential equation
.20and10if cos y y x y y
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Practice 3:1. Solve the differential equations by using Laplace transform,
where the function is subject to given conditions.a) d
b) F
c) F
2. Solve the differential equations when
by using Laplace transform,
20,02 y y y
99 y y
10,0 y y y
10,032 y y y
10,00 y y
Solution