590 likes | 890 Views
THE LAPLACE TRANSFORM. Chapter 4. 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 :
E N D
THE LAPLACE TRANSFORM Chapter 4
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.
Use the definition to find the values of the following: Example 1
Example2 Find the Laplace transform of the function
Transform of a Piecewise function Example 3 Given Find
Laplace Transform of a Derivative Let Find
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,
is a Linear Transform Where F and G are the transforms of some functions f and g .
Example 1 Solve the partial given IVP by Laplace transform.
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 Example 1: .
Exercise Solve
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 0 0 2 t -3 -3
The Second Translation Theorem If and then
Example 1 Let then
Example 2 Find where