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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?

5.1 Introduction

Laplace transforms will allow us to transform a Differential Equation into an algebraic equation.

5.1.1Definition and Notation

Let [pic], the Laplace transform of y is defined by

L[pic]

The transformed function is called [pic], thus

L[pic]

Revision: Integrating improper integrals

[pic]

[pic] is convergent if the limit is finite

and [pic] is divergent, if the limit is non-finite.

5.1.2Laplace Transforms of Some Simple Functions

The definition of the Laplace transform is

L [pic]

Finding Laplace transforms of some functions using basic principles.

Example 5.1 (a): Laplace transform of [pic]

If [pic], then L [pic]

[pic]

[pic]

( L {1} = [pic]

Since: [pic] then

L [pic] = [pic] L [pic]

Example 5.1 (b): Laplace transform of the form [pic]

Find L [pic].

L [pic] = 5 L [pic] = [pic]

Example 5.1 (c): Laplace transform of [pic]

If [pic], then

L [pic] L [pic]

[pic] - use integration by parts with

[pic]

[pic]

[pic]

← L [pic]

Example 5.1 (d): Laplace transform of [pic]

Find L [pic].

Therefore, L [pic][pic].

Using integration by parts twice:

[pic]

( L[pic][pic]

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:

(a) [pic] (b) [pic]

Solution

(a) [pic]

Looking at the table, we find

L[pic] =[pic]

So, L [pic]=[pic]

(b) [pic]

From the table, we find L {[pic]} = [pic]

Therefore, L {[pic]} = [pic]

Structured Examples: Finding Laplace Transforms

• Using the standard tables of Laplace Transforms

|Question 1 |Prompts/ Questions |

|Find the Laplace Transform of each of the following function: |What is a Laplace transform? |

|(a) [pic] |How does the function compares to the standard |

|(b) [pic] |function? |

| | |

| | |

How to find Laplace Transforms of [pic]?

We will need:

2. Properties of Laplace Transforms

1) Linearity

Let L [pic] and L [pic] exist with [pic] and [pic] constants, then

L [pic][pic] L [pic][pic] L [pic]

2) First Shift Theorem

Let L[pic] with a constant, then

L [pic]

Example 1.11 :Using tables and linearity law

(a) Find the Laplace Transform of each of the following function

(1) [pic] (2) [pic]

Solution

(1) [pic]

L [pic]= 2 L [pic]( 4 L [pic]+ L (1) = [pic] = [pic].

(2) [pic]

L [pic] = 2 L (sin 3t) ( L [pic]

= [pic].

Example 1.12: Using tables and the First Shift Theorem

(a) Find the Laplace Transform of each of the following function

(1) [pic] (2) [pic]

Solution

(1) L [pic] = [pic]

(2) L [pic] = [pic]

| | |

|Question 13 |Prompts/ Questions |

|Find the Laplace Transform of each of the following function: |What is a Laplace transform? |

|(a) [pic] |How does the function compares to the standard |

|(b) [pic] |function? |

|(c) [pic] |Which formula do you need to change the expression|

| |into the standard form? |

| |Which theorems must be used? |

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 [pic] L [pic]

If [pic] then L [pic] L [pic]

OR L [pic][pic]

Example 1.12 (a): Laplace transform of the first derivative

Find the Laplace Transform of [pic] with the initial condition that [pic].

Solution:

• Take Laplace transform of both sides

L [pic] = L [pic]

• Use Table of Laplace Transform

L [pic] L [pic]

L [pic]

s L [pic] = [pic]

s L [pic] = [pic].

L [pic]= [pic]

2) Transform of the Second Derivative

L [pic] L [pic]

If [pic] then L [pic] L [pic]

OR L [pic]

Example 1.12(b): Laplace transforms of the second derivative

Find the Laplace Transform of [pic] with the initial condition that [pic] and [pic]

Solution

From the tables: L [pic] L [pic] and L [pic] L [pic]

L [pic]= L (0)

We know that L [pic]= Y(s) and [pic] and [pic]

[pic]

[pic]

|Question 14 |Prompts/ Questions |

|Find the Laplace transform of each expression and substitute the given initial |What is the Laplace Transform of first and second |

|conditions |order derivatives? |

|(2) [pic] |What do you do with the initial values? |

| | |

|4) [pic] | |

2. Inverse Transforms

If L [pic], then [pic] is called the Inverse Laplace Transform of [pic] and is written as

L (1[pic][pic]

➢ 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.

|L (1[pic][pic] |

|[pic] |[pic] |

|[pic] |a |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

Example 1.13: Finding the inverse Laplace Transforms

Find [pic] for the following Laplace transforms:

(a) [pic] (b) [pic]

(c) [pic]

Solution:

(a) [pic]

Use the Table of Laplace transform: [pic]

(b) [pic] ( [pic]

(c) [pic]

➢ Rearrange or modify so that it looks like a function in the table

➢ make sure that you still have the original function:

[pic]

Review: Changing expressions into standard forms – some algebraic manipulations

1) Completing the square

2) Partial Fractions

Example 1.14(a): Completing the square

Find y if L [pic][pic]

Solution:

1) change [pic] 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?

[pic]

• Check numerator: modify if necessary

[pic]

[pic][pic][pic]

2) Find y

([pic][pic]

[pic]

Example 1.14 (b): Using Partial Fractions

Find y if ([pic][pic]

Solution:

1) modify to be comparable to standard forms in the tables:

• Check denominator: [pic]

• Check expression: suitable for conversion to partial fractions

[pic]

[pic]

Solve for A and B : Compare numerator:

[pic]

[pic]

Solve simultaneously:

[pic]

[pic]

[pic]

2) Find y

([pic]= ( [pic]

[pic]

REVISION:

Other Standard forms of partial fractions

(a) Expressions are of the form [pic] where [pic] and [pic] are polynomials in s and the degree of [pic] is less than the degree of [pic].

Example 1.15 (a): [pic] is a constant and [pic] has linear factors

[pic]

Example 1.15(b): [pic] is a constant and [pic] has linear factors with some factors repeated

[pic]

Example 1.15(c): [pic] is a constant and [pic] has linear and quadratic factors

(i) [pic]

(ii) [pic]

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 [pic] can be obtained from the inverse Laplace Transforms.

Example 1.16: Solve the differential equation [pic] with the initial conditions [pic]

Solution:

1. Take Laplace transforms of both sides

([pic] ([0]

( [[pic]]+ ([pic] ( ([pic] = 0

Use the formula of Laplace Transforms of first and second order derivatives:

[pic]([pic] + 2s([pic] (3([pic]

2. Solve for ([pic]

Given: [pic]; substitute the initial conditions

[pic]([pic] + 2s([pic] (3([pic]

([pic] [pic]

([pic][pic]

3. Convert into standard forms

Use partial fractions

[pic]

[pic]

Compare numerator:

[pic]

Thus,

[pic]

Solving simultaneously:

[pic] and [pic]

([pic] = [pic]

4. Find the inverse Laplace Transforms: use tables

[pic]

Example 1.17: Solve the differential equation [pic] where y is a function of t, if [pic] and [pic] are both zero at [pic].

Solution:

5. Take Laplace transforms of both sides

([pic] ([0]

([pic] + ([pic] ( ([pic] = 0

[pic]([pic] + 4([pic] ( [pic] = 0

6. Solve for ([pic]

[pic]([pic] = [pic] since [pic]

([pic] = [pic]

7. Convert into standard forms

Use partial fractions

[pic]

[pic]

Compare numerator:

[pic]

Thus,

[pic]

[pic]

([pic] = [pic]

8. Find the inverse Laplace Transform: use tables

[pic]

|Question 15 |Prompts/ Questions |

|Solve the following initial value problems |What is the Laplace Transforms or equations? |

|(a) [pic] |What is the Laplace Transform of first and second |

|(b) [pic] |order derivatives? |

| |What do you do with the initial values? |

|(c) [pic] |How does your expression compare to the standard |

| |forms? |

| |Which algebraic manipulations do you need? |

| |How do you find the inverse Laplace Transforms? |

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