CHAPTER 5

Laplace Transform

Focus of Attention

What is a Laplace Transforms?

How to use the standard Laplace

Transforms Tables?

What are the properties of Laplace

transforms? Linearity?

What are the First Shift Theorem?

Multiplication by t theorem?

What are the Laplace Transforms of

first and second order derivatives?

How to solve differential equations

using Laplace Transforms?

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5.1 Introduction

Laplace transforms will allow us to transform

a Differential Equation into an algebraic

equation.

5.1.1Definition and Notation

Let , the Laplace transform of y is defined

by

LThe transformed function is called , thus

L

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Revision: Integrating improper integrals

is convergent if the limit is finite

and is divergent, if the limit is non-

finite.

5.1.2Laplace Transforms of Some Simple FunctionsThe definition of the Laplace transform is

L

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Finding Laplace transforms of some functions

using basic principles.

Example 5.1 (a): Laplace transform of

If , then L

L {1} =

Since: then

L = L

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Example 5.1 (b): Laplace transform of the

form

Find L .

L = 5 L =

Example 5.1 (c): Laplace transform of

If , then

L L

- use integration by parts with

L

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Example 5.1 (d): Laplace transform of

Find L .

Therefore, L .

Using integration by parts twice:

L

Important note: must know how to use the

Tables of Laplace Transforms.

Example 5.2: Using tables to find Laplace

Transforms

Find the Laplace Transform of each of the

following function:

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(a) (b)

Solution

(a)

Looking at the table, we find

L =

So, L =

(b)

From the table, we find L { } =

Therefore, L { } =

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Structured Examples: Finding Laplace Transforms

Using the standard tables of Laplace

Transforms

Question 1Find the Laplace Transform

of each of the following

function:

(a)

(b)

Prompts/ Questions What is a

Laplace transform?

How does the function compares to the standard function?

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How to find Laplace Transforms of

?

We will need:

5.2 Properties of Laplace Transforms

(1) Linearity

Let L and L exist with and

constants, then

L L L

(2) First Shift Theorem

Let L with a constant, then

L

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Example 1.11 :Using tables and linearity

law

(a) Find the Laplace Transform of each of the

following function

(1) (2)

Solution

(1)

L = 2 L 4 L + L (1) =

= .

(2)

L = 2 L (sin 3t) L

= .

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Example 1.12: Using tables and the First

Shift Theorem

(a) Find the Laplace Transform of each of the

following function

(1) (2)

Solution

(1) L =

(2) L =

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Question 13Find the Laplace Transform

of each of the following

function:

(a)

(b)

(c)

Prompts/ Questions What is a

Laplace transform?

How does the function compares to the standard function?oWhich

formula do you need to change the expression into the standard form?

Which theorems must be used?

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1.3.1 Transform of a Derivative

We can find the Laplace transforms of the

first and second derivative. Thus, we can use

these transforms to convert the differential

equations into an algebraic form.

(1) Transform of the First Derivative

L L

If then L L

OR L

Example 1.12 (a): Laplace transform of the

first derivative

Find the Laplace Transform of with the

initial condition that .

Solution:

Take Laplace transform of both sides

L = L

Use Table of Laplace Transform

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L L

L

s L =

s L = .

L =

(2) Transform of the Second Derivative

L L

If then L L

OR L

Example 1.12(b): Laplace transforms of

the second derivative

Find the Laplace Transform of

with the initial condition that and

Solution

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From the tables: L L and L L

L = L (0)

We know that L = Y(s) and and

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Question 14Find the Laplace transform

of each expression and

substitute the given initial

conditions

(2)

4)

Prompts/ Questions What is the

Laplace Transform of first and second order derivatives?o What do

you do with the initial values?

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1.3.2 Inverse Transforms

If L , then is called the Inverse

Laplace Transform of and is written as

L 1

We find the inverse Laplace

Transform by reading the same tables but

in reverse.

We have chosen a few functions to

illustrate how it is done.

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L 1

a

Example 1.13: Finding the inverse Laplace

Transforms

Find for the following Laplace

transforms:

(a) (b)

(c)

Solution:

(a) 59

Use the Table of Laplace transform:

(b)

(c)

Rearrange or modify so that it looks

like a function in the table

make sure that you still have the

original function:

Review: Changing expressions into

standard forms – some algebraic

manipulations

(1) Completing the square

(2) Partial Fractions

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Example 1.14(a): Completing the square

Find y if L

Solution:

1) change to an expression in a

form similar to those that can be found in

the tables.

Check denominator: standard? factorise?

use method of completing the square?

Check numerator: modify if necessary

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2) Find y

Example 1.14 (b): Using Partial Fractions

Find y if

Solution:

1) modify to be comparable to standard

forms in the tables:

Check denominator:

Check expression: suitable for conversion

to partial fractions

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Solve for A and B : Compare numerator:

Solve simultaneously:

2) Find y

=

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REVISION:

Other Standard forms of partial fractions

(a) Expressions are of the form where

and are polynomials in s and the

degree of is less than the degree of .

Example 1.15 (a): is a constant and

has linear factors

Example 1.15(b): is a constant and

has linear factors with some factors

repeated

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Example 1.15(c): is a constant and

has linear and quadratic factors

(i)

(ii)

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1.3.3 Solving Differential Equations using

Laplace Transform

The method converts the differential

equations into algebraic expressions in

terms of s.

The expressions have to be

manipulated such that the function

can be obtained from the inverse

Laplace Transforms.

Example 1.16: Solve the differential equation

with the initial conditions

Solution:

1. Take Laplace transforms of both sides

[0]

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[ ]+ = 0

Use the formula of Laplace Transforms of

first and second order derivatives:

+ 2s 3

2. Solve for

Given: ; substitute the

initial conditions

+ 2s 3

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3. Convert into standard forms

Use partial fractions

Compare numerator:

Thus,

Solving simultaneously:

and

=

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4. Find the inverse Laplace Transforms:

use tables

Example 1.17: Solve the differential equation

where y is a function of t, if

and are both zero at .

Solution:

5. Take Laplace transforms of both sides

[0]

+ = 0

+ 4 = 0

6. Solve for

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= since

=

7. Convert into standard forms

Use partial fractions

Compare numerator:

Thus,

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=

8. Find the inverse Laplace Transform:

use tables

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Question 15Solve the following initial value problems

(a)

(b)

(c)

Prompts/ Questions What is the

Laplace Transforms or equations?

What is the Laplace Transform of first and second order derivatives?oWhat do you do

with the initial values?

How does your expression compare to the standard forms?o Which

algebraic manipulations do you need?

How do you find the inverse Laplace Transforms?

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