Chapter 4 THE LAPLACE TRANSFORM

Plan I - Definition and basic properties II - Inverse Laplace transform and solutions of DE III - Operational Properties

I – Definitions and basic properties Learning objective At the end of the lesson you should be able to : Define Laplace Transform. Find the Laplace Transform of different type of functions using the definition.

Definition: Laplace Transform Let f be a function defined for Then the integral is said to be the Laplace transform of f, provided that the integral converges.

Notations

Use the definition to find the values of the following: Example 1 Use the definition to find the values of the following:

Solution

Solution

Theorem: Transforms of some Basic Functions

is a Linear Transform

Find the Laplace transform of the function Example2 Find the Laplace transform of the function

Example2

Transform of a Piecewise function Example 3 Given Find

Solution

Laplace Transform of a Derivative Let Find

Laplace Transform of a Derivative

Laplace Transform of a Derivative Theorem where

Laplace Transform of a Derivative Example Find the Laplace transform of the following IVP

Laplace Transform of a Derivative Solution

Laplace Transform of a Derivative Solution

II – Inverse Laplace Transform and solutions of DEs Learning objective At the end of the lesson you should be able to : Define Inverse Laplace Transform. Solve ODEs using the Laplace Transform.

Inverse Transforms If F (s) represents the Laplace transform of a function f (t), i.e., L {f (t)}=F (s) then f (t) is the inverse Laplace transform of F (s) and,

Theorem : Some Inverse Transforms

Application

is a Linear Transform Where F and G are the transforms of some functions f and g .

Division and Linearity Find

Division and Linearity

Partial Fractions in Inverse Laplace Find

Partial Fractions in Inverse Laplace ,

Solve the partial given IVP by Laplace transform. Example 1 Solve the partial given IVP by Laplace transform.

Solution 1

Solution 1

Solution 1

III – Operational Properties Learning objective At the end of the lesson you should be able to use translation theorems.

First translation theorem If and is any real number, then .

First translation theorem Example 1: .

First translation theorem Example 2: .

Inverse form of First translation theorem .

Inverse form of First translation theorem Example 1: .

Exercise Solve

Solution

Solution

Solution

Solution

Unit Step Function or Heaviside Function The unit step function is defined as U 1 t

Example What happen when is multiplied by the Heaviside function

Example f f t 2 t -3 -3

The Second Translation Theorem If and then

Example 1 Let then

Example 2 Find where

Example 2 For We would like to apply the previous theorem So, we write

Example 2 Then where Finally

The Inverse Second Translation Theorem If then

Example Evaluate

Example therefore,

Summary .