1. Represented by Dr. Shorouk Ossama
(Laplace Transform)
Represented by
Dr. Shorouk Ossama

2. Definition
We are going to study of transforming differential equations into algebraic equations.
Laplace Transform L for the function f (t) is defined by:

3. Example:
Find Laplace Transform For f (t):
Solution
a)The Laplace transform for f (t) = 1 given by:
a)The Laplace transform for f (t) = t given by:

4. In general

5. Example:
Find Laplace Transform For: 𝒆𝒂𝒕 , sin at , cos at, sinh at, cosh at
Solution

8. Theorem (1): Linearity
The Laplace Transform has these inherited integral properties:
a) L{ f (t) + g (t) } = L{ f (t)} + L {g (t) }
b) L{ c f (t) } = L c { f (t) }
Example:
Let f (t) = t (t – 1) – sin 2t + e3t , compute L{f (t)}
Solution
L {f (t)} = L { t2 – t – sin 2t + e3t } =
L {t2} –L{t} –L{sin 2t} + L {e3t } = 2 𝑠 3 − 1 𝑠 2 − 2 𝑠 𝑠−3

9. Example:
Find The Laplace Transform For The Following
(a) f (t) = 3 t2 + sin 5t

10. 𝟏 𝟐 L { 1 – cos 4t }

11. Theorem (2): The s- Differential Rule
Let f (t) be of exponential order, then
L{ t f (t) } = - 𝒅 𝒅𝒔 𝑳 𝒇 𝒕
L{ 𝒕 𝒏 f (t) } = (−𝟏) 𝒏 𝒅 𝒏 𝒅 𝒔 𝒏 𝑳 𝒇 𝒕
Example: Find The Following
(a) L {t cos at }
𝑮𝒆𝒏𝒆𝒓𝒂𝒍

13. Theorem (3): First Shifting Rule
L{ 𝒆𝒂𝒕 f (t) } = 𝑭 (𝒔−𝒂)
Example:
Find The Following
L { 𝒆𝒂𝒕 cos ⍵t }
(d) L { 𝒕 𝒆𝒂𝒕 sin ⍵t }

14. 3 𝑠 2 +9
𝑠 𝑠 2 +25

15. L{ 𝒕 𝒏 f (t) } = (−𝟏) 𝒏 𝒅 𝒏 𝒅 𝒔 𝒏 𝑳 𝒇 𝒕
Summary
L{ 𝒕 𝒏 f (t) } = (−𝟏) 𝒏 𝒅 𝒏 𝒅 𝒔 𝒏 𝑳 𝒇 𝒕
n = …..
L f t
Get (−𝟏) 𝒏 𝒅 𝒏 𝒅 𝒔 𝒏
L{ 𝒆𝒂𝒕 f (t) } = 𝑭 (𝒔−𝒂)
a = …..
𝑠 →𝑠−𝑎

17. SUMMARY
From Pages 70 To 77

18. Thanks