Scribd
Upload
85 views2 pages

Laplace Transform

1. The document defines the Laplace transform and provides the Laplace transforms of several common functions, including constants, exponentials, polynomials, sines, cosines, hyperbolic sines and hyperbolic cosines. 2. It also covers the linearity of the Laplace transform and provides the first shifting theorem. Examples are given of taking the Laplace transform of functions involving exponentials and trigonometric functions. 3. Formulas are provided for sum and product identities of sines and cosines, as well as half angle and double angle identities that can be used to evaluate Laplace transforms.

Uploaded by

lloydalejandria13
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
85 views2 pages

Laplace Transform

1. The document defines the Laplace transform and provides the Laplace transforms of several common functions, including constants, exponentials, polynomials, sines, cosines, hyperbolic sines and hyperbolic cosines. 2. It also covers the linearity of the Laplace transform and provides the first shifting theorem. Examples are given of taking the Laplace transform of functions involving exponentials and trigonometric functions. 3. Formulas are provided for sum and product identities of sines and cosines, as well as half angle and double angle identities that can be used to evaluate Laplace transforms.

Uploaded by

lloydalejandria13
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 2

LAPLACE TRANSFORM 3.

𝑓(𝑡) = 𝑐𝑜𝑠4𝑡
Definition : the laplace transform of a
function 𝑓(𝑡) is transformation from the
function of a variable 𝑡 to a function of
variable 𝑠 denoted by 4. 𝑓(𝑡) = 𝑠𝑖𝑛4𝑡

ℒ {𝐹(𝑡)} or 𝐹(𝑠)
∞
ℒ {𝐹(𝑡)} or 𝐹(𝑠) = ∫0 𝑒 −𝑠𝑡 𝑓(𝑡)𝑑𝑡
5. 𝑓(𝑡) = 𝑠𝑖𝑛ℎ4𝑡

Laplace Transforms of the common


functions
1
𝑘
6. 𝑓(𝑡) = cosh 4 𝑡
ℒ{𝑘} =
𝑠
1
ℒ{𝑒 𝑘𝑡 } = 𝑠−𝑘
7. 𝑓(𝑡) = 4𝑠𝑖𝑛8𝑡 − 5𝑐𝑜𝑠3𝑡
𝑛!
ℒ{𝑡 𝑛 } = ; 𝑛 >0
𝑠 𝑛+1
𝑠
ℒ{𝑐𝑜𝑠𝑘𝑡} = 𝑠2+ 𝑘2
𝑘
ℒ{𝑠𝑖𝑛𝑘𝑡} = 𝑡4
𝑠2+ 𝑘2 8. 𝑓(𝑡) = + 3𝑠𝑖𝑛8𝑡
36
𝑠
ℒ{𝑐𝑜𝑠ℎ𝑘𝑡} = 𝑠2− 𝑘2
𝑘
ℒ{𝑠𝑖𝑛ℎ𝑘𝑡} = 𝑠2− 𝑘2

9. 𝑓(𝑡) = 3𝑠𝑖𝑛(𝜔𝑡 + 𝛼)
Linearity of laplace transform
1. ℒ{𝑐 𝑓(𝑡)} = 𝑐 ℒ{𝑓(𝑡)}
2. ℒ{ 𝑓(𝑡) ± 𝑔(𝑡)} =
ℒ{𝑓(𝑡)} ± ℒ{ 𝑔(𝑡)} 10. 𝑓(𝑡) = 𝑠𝑖𝑛3𝑡𝑐𝑜𝑠3𝑡 + 3

Example :
Solve the laplace the following .
1. 𝑓(𝑡) = 𝑡 3 11. 𝑓(𝑡) = 𝑠𝑖𝑛3𝑡𝑠𝑖𝑛4𝑡

2. 𝑓(𝑡) = 𝑒 4𝑡
12. 𝑓(𝑡) = 𝑠𝑖𝑛2 4𝑡

Page 1
13. 𝑓(𝑡) = 𝑠𝑖𝑛3𝑡𝑐𝑜𝑠3𝑡

Recall : First Shifting Theorem


Sum and product of sine and cosine Identities Theorem :

1 If ℒ{𝑓(𝑡)} = 𝐹(𝑠) then


𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦 = [𝑐𝑜𝑠(𝑥 − 𝑦) − 𝑐𝑜𝑠(𝑥 + 𝑦)]
2
ℒ{𝑒 𝑎𝑡 𝑓(𝑡)} = 𝐹(𝑠 − 𝑎)
1
𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑦 = [𝑠𝑖𝑛(𝑥 − 𝑦) + 𝑠𝑖𝑛(𝑥 + 𝑦)]
2
1
𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 = 2
[𝑐𝑜𝑠(𝑥 − 𝑦) − 𝑐𝑜𝑠(𝑥 + 𝑦)] Example :
𝑠𝑖𝑛(𝑥 + 𝑦 ) = 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑦 + 𝑐𝑜𝑠𝑥𝑠𝑖𝑛𝑦 Evaluate the laplace of the following
𝑠𝑖𝑛(𝑥 − 𝑦 ) = 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑦 − 𝑐𝑜𝑠𝑥𝑠𝑖𝑛𝑦 1. 𝑓(𝑡) = {𝑒 𝑎𝑡 𝑡 𝑛 }
𝑐𝑜𝑠(𝑥 + 𝑦 ) = 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 − 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦
𝑐𝑜𝑠(𝑥 − 𝑦 ) = 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 + 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦

Double Identities
2. 𝑓(𝑡) = {𝑒 −4𝑡 𝑐𝑜𝑠3𝑡}
𝑠𝑖𝑛2𝜃 = 2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃
𝑐𝑜𝑠2𝜃 = 𝑐𝑜𝑠 2 𝜃 − 𝑠𝑖𝑛2 𝜃
2𝑡𝑎𝑛𝜃
𝑡𝑎𝑛2𝜃 =
1 − 𝑡𝑎𝑛2 𝜃

Half Angle Identities


3. 𝑓(𝑡) = {3𝑠𝑖𝑛2𝑡 + 5𝑐𝑜𝑠8𝑡}
1−𝑐𝑜𝑠2𝜃
𝑠𝑖𝑛2 𝜃 =
2
1−𝑐𝑜𝑠2𝜃
𝑐𝑜𝑠 2 𝜃 = 2

𝜃 1−𝑐𝑜𝑠𝜃
𝑠𝑖𝑛 2 = ±√ 2 4. 𝑓(𝑡) = {𝑒 2𝑡 𝑠𝑖𝑛𝑗2𝑡}
𝜃 1+𝑐𝑜𝑠𝜃
𝑐𝑜𝑠 2 = ±√ 2

𝜃 1 − 𝑐𝑜𝑠𝜃
𝑡𝑎𝑛 2 = ±√ 1+ 𝑐𝑜𝑠𝜃

5. 𝑓(𝑡) = {2𝑒 2𝑡 𝑠𝑖𝑛4𝑡𝑠𝑖𝑛3𝑡 𝑧}

7 0<𝑡<3
6. 𝑓(𝑡) = { 2 3 < 𝑡 < 5}
0 𝑡≥4

Page 2

You might also like