1. Class Notes 12: Laplace Transform (1/3) Intro & Differential Equations
82 – Engineering Mathematics

2. Laplace Transform – Introduction

3. Laplace Transform – Introduction

4. Pierre Simon Laplace (Marquis)
French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) ( ). This seminal work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems.
He formulated Laplace's equation, and invented the Laplace transform. In mathematics, the Laplace transform is one of the best known and most widely used integral transforms. It is commonly used to produce an easily solvable algebraic equation from an ordinary differential equation. It has many important applications in mathematics, physics, optics, electrical engineering, control engineering, signal processing, and probability theory.
He restated and developed the nebular hypothesis of the origin of the solar system and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse.
Pierre-Simon, marquis de Laplace
23 April March 1827
Video Clip

5. Laplace Transform – Motivation
Transform between the time domain (t) to the frequency domain (s)
Lossless Transform – No information is lost when the transform and the inverse transform is applied
Frequency
Domain (S)
Time
Domain (t)
Differential
Equation
Algebraic
Equation
Time Domain
Solution
Frequency Domain
Solution

6. Improper Integral – Definition
An improper integral over an unbounded interval is defined as the limit of an integral over a finite interval
where A is a positive real number. If
the integral from a to A exists for each A > a
the limit as A → ∞ exists
then
the improper integral is said to converge to that limiting value.
Otherwise, the integral is said to diverge or fail to exist.

7. Improper Integral – Example

8. Integral Transform - Laplace Transform – Definition
Tool for solving linear diff. eq. – Integral transform
k(s,t) - The kernel of the transformation
Transform
Laplace transform
whenever this improper integral converges

9. Suppose that f is a function for which the following hold:
Laplace Transform – Theorem – Sufficient Conditions for Existence
Suppose that f is a function for which the following hold:
(1) f is piecewise continuous on [0, b] for all b > 0.
(2) | f(t) |  Meat when t  T, with T, M > 0.
Then the Laplace Transform of f exists for s > a.
Note: A function f that satisfies the conditions specified above is said to have exponential order as t  .

10. a = t0 < t1 < … < tn = b such that
Laplace Transform – Theorem – Condition No. 1 - Piecewise Continuous
A function f is piecewise continuous on an interval [a, b] if this interval can be partitioned by a finite number of points
a = t0 < t1 < … < tn = b such that
(1) f is continuous on each (tk, tk+1)
In other words, f is piecewise continuous on [a, b] if it is continuous there except for a finite number of jump discontinuities.

11. Laplace Transform – Theorem 6. 1. 2 – Condition No
Laplace Transform – Theorem – Condition No. 1 - Piecewise Continuous

12. Laplace Transform – Theorem 6. 1. 2 – Condition No
Laplace Transform – Theorem – Condition No. 2 - Exponential Order If
f is an increasing function
Then
| f(t) |  Meat when t  T, with T, M > 0.

13. Laplace Transform – Theorem 6. 1. 2 – Condition No
Laplace Transform – Theorem – Condition No. 2 - Exponential Order
A positive integral power of t is always of exponential order since for c>0

14. Laplace Transform (Function) – Example

15. Laplace Transform (Function) – Example

16. Laplace Transform (Function) – Example

17. Laplace Transform (Function) – Example
second integration by parts

18. Laplace Transform – Discontinues Function - Example
Transformation of piecewise continuous function

19. Operation Properties – Translation on the t-Axis (Time) Second Translation Theorem

20. Operation Properties – Translation on the S-Axis (Freq
Operation Properties – Translation on the S-Axis (Freq.) First Translation Theorem
Example:

21. Laplace Transform – Linearity
Suppose f and g are functions whose Laplace transforms exist for s > a1 and s > a2, respectively.
Then, for s greater than the maximum of a1 and a2, the Laplace transform of c1 f (t) + c2g(t) exists. That is,
with

22. Laplace Transform – Linearity – Example

24. Inverse Laplace Transform – Example

26. Inverse Laplace Transform – Example

28. Inverse Laplace Transform – Example
(linearity)

29. Inverse Laplace Transform – Partial Fraction
Option 1: 3 equations with 3 unknown variables A, B, C
Option 2: Plug (s=1) 16=A(-1)(5) A=-16/5
Plug (s=2) 25=B(1)(6) B=25/6
Plug (s=-4) 1=C(-5)(-6) C=1/30

30. Inverse Laplace Transform – Partial Fraction

31. Laplace Transform of a Derivative (First)

32. Laplace Transform of a Derivative (Second)

33. Laplace Transform of a Derivative
first derivative
Initial Conditions
second derivative
third derivative
n-th derivative

34. Laplace Transform – Solving Linear ODEs
Initial Conditions:
- constants
(I.C.s) - constants
Apply Laplace Transform

35. Laplace Transform – Solving Linear ODEs
The Laplace Transform of a linear differential equation with constant coefficients become an algebraic equation in Y(s)
Time Domain
Differential Equation
Frequency Domain
Algebraic Equation
Laplace Transform

36. Laplace Transform – Example First Order Linear ODEs

37. Laplace Transform – Example First Order Linear ODEs

38. Laplace Transform – Example Second Order Linear ODEs
(See previous example)