Basic Laplace Transforms: Difference between revisions

From Class Wiki
Jump to navigation Jump to search
(New page: [[Image:[[Image:'''Basic Laplace Transforms''' The laplace transform has the standard form of: :<math>F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt </math> (C...)
 
No edit summary
 
(4 intermediate revisions by one other user not shown)
Line 1: Line 1:
[[Image:[[Image:'''Basic Laplace Transforms'''
'''Basic Laplace Transforms'''


The laplace transform has the standard form of:
The laplace transform has the standard form of:
Line 5: Line 5:
:<math>F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt </math> (Cited From Fullerton, Colby)
:<math>F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt </math> (Cited From Fullerton, Colby)


However, in this class applying the standard form exclusively to solve problems is not practical. The use of Laplace transform properties greatly simplifies problems. These properties are listed in the book on page 525. The simple properties are listed below and as imported images from mathcad.
However, in this class applying the standard form exclusively to solve problems is not practical. The use of Laplace transform properties greatly simplifies problems. These properties are listed in the book on page 525. The simple properties are listed below and are imported images from mathcad.


'''Linearity'''
'''Linearity'''
Line 15: Line 15:




'''Examples:'''
'''Example:'''

If F(s)=(s+2)/(S+1) and f(t)=0 for t<0, then find the Laplace transform for the following functions identifying each property used to compute answers.

'''(a)''' <math>g_1(t)=5f(t-2)</math> '''(b)'''<math>g_2(t)=5(e^{-2t})f(t)</math>



(a) Linearity:
<math>F(s)=(5(s+2))/(s+1)</math>

Time Shift:
<math>\mathcal{L} \left\{f(t-2)u(t-2)\right\}=e^{-2s}F(s) </math> = '''<math>(5e^{-2s}(s+2))/(s+1)</math>'''


(b) Linearity: <math>a_1F(s) = {5(s+2)}/(s+1) </math>
Frequency Shift:
<math>\mathcal{L} \left\{e^{-at}f(t)\right\}=F(s+a) </math>

<math>(5((s+2)+2)))/((s+2)+1)=(5s+20)/(s+1)</math>


<u>'''Author:'''</u>
Jaymin Joseph

'''<u>Reviewed by:</u>'''
*[[Grant, Joshua|Joshua Grant]]



<u>'''Read By:'''</u>

Latest revision as of 12:40, 20 January 2010

Basic Laplace Transforms

The laplace transform has the standard form of:

(Cited From Fullerton, Colby)

However, in this class applying the standard form exclusively to solve problems is not practical. The use of Laplace transform properties greatly simplifies problems. These properties are listed in the book on page 525. The simple properties are listed below and are imported images from mathcad.

Linearity

P1.jpg


P2.jpg


Example:

If F(s)=(s+2)/(S+1) and f(t)=0 for t<0, then find the Laplace transform for the following functions identifying each property used to compute answers.

(a) (b)


(a) Linearity:

Time Shift: =


(b) Linearity:

    Frequency Shift:


Author: Jaymin Joseph

Reviewed by:


Read By: