1. Chapter 4
THE LAPLACE TRANSFORM

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

3. 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.

4. 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.

5. Notations

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

7. Solution

8. Solution

9. Theorem: Transforms of some Basic Functions

10. is a Linear Transform

11. Find the Laplace transform of the function
Example2
Find the Laplace transform of the function

12. Example2

13. Transform of a Piecewise function
Example 3
Given
Find

14. Solution

15. Laplace Transform of a Derivative
Let
Find

16. Laplace Transform of a Derivative

17. Laplace Transform of a Derivative
Theorem
where

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

19. Laplace Transform of a Derivative
Solution

20. Laplace Transform of a Derivative
Solution

21. 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.

22. 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,

23. Theorem : Some Inverse Transforms

24. Application

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

26. Division and Linearity
Find

27. Division and Linearity

28. Partial Fractions in Inverse Laplace
Find

29. Partial Fractions in Inverse Laplace,

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

31. Solution 1

32. Solution 1

33. Solution 1

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

35. First translation theoremIf
and
is any real number, then .

36. First translation theorem
Example 1: .

37. First translation theorem
Example 2: .

38. Inverse form of First translation theorem.

39. Inverse form of First translation theorem
Example 1: .

40. Exercise
Solve

41. Solution

42. Solution

43. Solution

44. Solution

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

46. Example
What happen when is
multiplied by the Heaviside function

47. Example f f t 2 t -3 -3

48. The Second Translation Theorem
If and then

49. Example 1
Let
then

50. Example 2
Find
where

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

52. Example 2
Then
where
Finally

53. The Inverse Second Translation Theorem
If then

54. Example
Evaluate

55. Example
therefore,

56. Summary .