3 - Laplace Transform

8/12/2019 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



4

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



Transform

Inverse

Transform

Inverse



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



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



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



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)



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.



1515

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


First

derivative

Step 1


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


Secondderivative

Step 1

Transform fromthe table of LP

: F10




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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

3 - Laplace Transform

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