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Inverse Laplace Transform Examples, Lecture notes of Engineering

University of Bohol (UB)Engineering

Inverse Laplace Transform Examples

Typology: Lecture notes

2022/2023

Uploaded on 05/04/2023

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Inverse Laplace Transform
If ๐ฟ{๐‘“(๐‘ก)} = ๐น(๐‘ ), then ๐ฟโˆ’1{๐น(๐‘ )} = ๐‘“(๐‘ก)
๐‘ญ(๐’”) ------------------------------------------------------------- ๐ฟโˆ’1{๐น(๐‘ )} = ๐’‡(๐’•)
๐’‚
๐’” ------------------------------------------------------------- ๐’‚
๐Ÿ
๐’”โˆ’๐’Œ ------------------------------------------------------------- ๐’†๐’Œ๐’•
๐’Œ
๐’”๐Ÿ+๐’Œ๐Ÿ ------------------------------------------------------------- ๐ฌ๐ข๐ง๐’Œ๐’•
๐’”
๐’”๐Ÿ+๐’Œ๐Ÿ ------------------------------------------------------------- ๐œ๐จ๐ฌ๐’Œ๐’•
๐’Œ
๐’”๐Ÿโˆ’๐’Œ๐Ÿ ------------------------------------------------------------- ๐ฌ๐ข๐ง๐ก๐’Œ๐’•
๐’”
๐’”๐Ÿโˆ’๐’Œ๐Ÿ ------------------------------------------------------------- ๐œ๐จ๐ฌ๐ก๐’Œ๐’•
๐Ÿ
๐’”๐’+๐Ÿ ------------------------------------------------------------- ๐’•๐’
๐’!
From Theorem I:
๐‘ญ(๐’” ยฑ ๐’‚) ----------------------------------------------------------- ๐’†โˆ“๐’‚๐’•๐‘ณโˆ’๐Ÿ{๐‘ญ(๐’”)}
Example:
1. ๐ฟโˆ’1 {๐‘ 
๐‘ 2โˆ’9}
๐ฟโˆ’1 {๐‘ 
๐‘ 2โˆ’9}= ๐ฟโˆ’1 {๐‘ 
๐‘ 2โˆ’(3)2}
๐ฟโˆ’1 {๐‘ 
๐‘ 2โˆ’9}= cosh3๐‘ก
2. ๐ฟโˆ’1 {15
๐‘ 2+9}
๐ฟโˆ’1 {15
๐‘ 2+9}= ๐ฟโˆ’1 {5(3)
๐‘ 2+(3)2}
๐ฟโˆ’1 {15
๐‘ 2+9}= 5๐ฟโˆ’1 {3
๐‘ 2+(3)2}
๐ฟโˆ’1 {15
๐‘ 2+9}= 5sin3๐‘ก
pf3

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Inverse Laplace Transform

If ๐ฟ{๐‘“(๐‘ก)} = ๐น(๐‘ ), then ๐ฟโˆ’1{๐น(๐‘ )} = ๐‘“(๐‘ก)

๐’” -------------------------------------------------------------^ ๐’‚

๐’”โˆ’๐’Œ -------------------------------------------------------------^ ๐’†

๐’”๐Ÿ+๐’Œ๐Ÿ^ -------------------------------------------------------------^ ๐ฌ๐ข๐ง ๐’Œ๐’•

๐’”๐Ÿ+๐’Œ๐Ÿ^ -------------------------------------------------------------^ ๐œ๐จ๐ฌ ๐’Œ๐’•

๐’”๐Ÿโˆ’๐’Œ๐Ÿ^ -------------------------------------------------------------^ ๐ฌ๐ข๐ง๐ก ๐’Œ๐’•

๐’”๐Ÿโˆ’๐’Œ๐Ÿ^ -------------------------------------------------------------^ ๐œ๐จ๐ฌ๐ก ๐’Œ๐’•

๐’”๐’+๐Ÿ^ -------------------------------------------------------------^

From Theorem I:

Example:

1. ๐ฟโˆ’1^ {

๐‘ ^2 โˆ’9}

๐ฟโˆ’1^ {

๐‘ ^2 โˆ’9} = ๐ฟ

๐‘ ^2 โˆ’(3)^2 }

๐ฟโˆ’1^ {

๐‘ ^2 โˆ’9} = cosh 3๐‘ก

2. ๐ฟโˆ’1^ {

๐‘ ^2 +9}

๐ฟโˆ’1^ {

๐‘ ^2 +9} = ๐ฟ

๐‘ ^2 +(3)^2 }

๐ฟโˆ’1^ {

๐‘ ^2 +9} = 5๐ฟ

๐‘ ^2 +(3)^2 }

๐ฟโˆ’1^ {

๐‘ ^2 +9} = 5 sin 3๐‘ก

3. ๐ฟโˆ’1^ {

๐‘ ^4 }

๐ฟโˆ’1^ {

๐‘ ^4 } = ๐ฟ

๐ฟโˆ’1^ {

๐‘ ^4 } =^

๐‘ก^3

๐ฟโˆ’1^ {

๐‘ ^4 } =^

4. ๐ฟโˆ’1^ {

(๐‘ +4)^3 }

๐ฟโˆ’1^ {

(๐‘ +4)^3 } = ๐‘’

๐‘ ^3 }

๐ฟโˆ’1^ {

(๐‘ +4)^3 } = ๐‘’

๐ฟโˆ’1^ {

(๐‘ +4)^3 } = ๐‘’

โˆ’4๐‘ก ๐‘ก^2

๐ฟโˆ’1^ {

(๐‘ +4)^3 } =^