**Unit 5: Integral Transforms**
*Integral transforms form another class of tools that can be used to
convert linear differential equations into algebraic equations. An
integral transform is a mathematical operator that transforms a function
on which it operates into another function through integration over a
kernel. Often, by applying an integral transform, a complex function
can be represented as a sum of functions (usually known as special
functions) that are simpler to manipulate in the context of the
underlying ODE. This technique is closely related to the earlier
discussion of power series solutions.*

*A wide variety of integral transforms find application in the study of
linear differential equations. One useful feature of integral
transforms is spectral factorization, or the representation of an
arbitrary function as the sum of a series of orthogonal basis
functions. The differential equation is thereby transformed into an
algebraic problem, which can be solved using linear algebra. Once the
solution is obtained, the solution to the original problem can be found
by applying an inverse integral transform. *

**Unit5 Learning Outcomes**

Upon successful completion of this unit, the student will be able to:

- Describe how integral transforms can be used to reduce the
complexity of selected classes of mathematics problems.
- Find the solution for ordinary differential equations using integral
transforms.
- Identify discrete spectra associated with selected Laplace
transforms.
- Find the inverse Laplace transform.
- Find the solution for initial value problems using Laplace
transforms.

**5.1 Introduction to Integral Transforms**
- **Reading: Dublin City University: Eugene O’Riordan’s “MS227. Linear
Mathematics”: “Laplace Transform”**
Link: Dublin City University: Eugene O’Riordan’s “MS227. Linear
Mathematics”: “Laplace
Transform” (PDF)

Instructions: Click on the link above. Under the title “MS227
Linear Mathematics,” click on the link for “Laplace Transform.”
Read the entire document.

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displayed on the webpages above.

**5.2 Discrete Spectra - The Laplace Transform**
- **Reading: Lamar University: Paul Dawkins’ Differential Equations:
“Laplace Transforms”: “The Definition”**
Link: Lamar University: Paul Dawkins’ *Differential Equations*:
“Laplace Transforms”: “The
Definition”
(PDF)

Instructions: Click on the link above and then look for the line
that says “Here is the file you requested: Differential Equations
(Math 3301).” Click on the link associated with “Differential
Equations (Math 3301).” Then read the section titled: “The
Definition” on pages 183-186.

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displayed on the webpages above.

**Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Laplace Transforms”**Link: Lamar University: Paul Dawkins’*Differential Equations*: “Laplace Transforms”: “Laplace Transforms” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Go to Page 187 and read the section titled: “Laplace Transforms” on pages 187-190. Work on examples 1 and 2 on your own before looking at the solutions. After finishing with your work, look at the solutions.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3A. Elementary Properties and Formulas”**Link: MIT: Professor Arthur Mattuck’s*18.03 Notes and Exercises*: “Exercises”: “Laplace Transform”: “3A Elementary Properties and Formulas” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 3A-1, 3A-2, 3A-4, 3A-5, 3A-7, 3A-8, and 3A-9. When you finish, please check your answers to these exercises with “Section 3 Solutions.”

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**5.3 The Inverse Laplace Transform**
- **Reading: Lamar University: Paul Dawkins’ Differential Equations:
“Laplace Transforms”: “Inverse Laplace Transforms”**
Link: Lamar University: Paul Dawkins’ *Differential Equations*:
“Laplace Transforms”: “Inverse Laplace
Transforms”
(PDF)

Instructions: Click on the link above and then look for the line
that says “Here is the file you requested: Differential Equations
(Math 3301).” Click on the link associated with “Differential
Equations (Math 3301).” Read the section titled: “Inverse Laplace
Transforms” on pages 191 to 201. Work on examples 2 and 3 on your
own before looking at the solutions. After finishing with your
work, look at the solutions.

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displayed on the webpages above.

**Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Step Functions”**Link: Lamar University: Paul Dawkins’*Differential Equations*: “Laplace Transforms”: “Step Functions” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled: “Laplace Transforms” on pages 202 to 214. Work on examples 2 and 3 on your own before looking at the solutions. After finishing with your work, look at the solutions.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Dirac Delta Function”**Link: Lamar University: Paul Dawkins’*Differential Equations*: “Laplace Transforms”: “Dirac Delta Function” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Go to page 233 and read the “Dirac Delta Function” section.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Convolution Integrals”**Link: Lamar University: Paul Dawkins’*Differential Equations*: “Laplace Transforms”: “Convolution Integrals” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Go to Page 236. Read the section titled “Convolution Integrals,” beginning on page 236, and ending with example 1 on page 237. Then, go to page 239 and read pages 239 and 240. The table on page 239 will be used for the exercises, so make sure that you have it available when working with the assignments.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3A. Elementary Properties and Formulas”**Link: MIT: Professor Arthur Mattuck’s*18.03 Notes and Exercises*: “Exercises”: “Laplace Transform”: “3A Elementary Properties and Formulas” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 3A-3 and 3A-10. When you finish, please check your answers to these exercises with “Section 3 Solutions.”

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**5.4 Using Laplace Transforms to Solve Initial Value Problems**
- **Reading: Lamar University: Paul Dawkins’ Differential Equations:
“Laplace Transforms”: “Solving IVP’s with Laplace Transforms”**
Link: Lamar University: Paul Dawkins’ *Differential Equations*:
“Laplace Transforms”: “Solving IVP’s with Laplace
Transforms”
(PDF)

Instructions: Click on the link above and then look for the line
that says “Here is the file you requested: Differential Equations
(Math 3301).” Click on the link associated with “Differential
Equations (Math 3301).” Go to page 215 and read the section titled
“Solving IVP’s with Laplace Transforms” on pages 215 to 221. Work
on examples 2 and 3 on your own. After you finish, look at the
solutions in the reading.

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displayed on the webpages above.

**Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Non Constant Coefficients IVP’s”**Link: Lamar University: Paul Dawkins’*Differential Equations*: “Laplace Transforms”: “Non Constant Coefficients IVP’s” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Non Constant Coefficients IVP’s” on pages 222 to 225. Work on example 2 on your own. After you finish, look at the solution provided in the reading.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “IVP’s with Steps Functions”**Link: Lamar University: Paul Dawkins’*Differential Equations*: “Laplace Transforms”: “IVP’s with Steps Functions” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “IVP’s with Steps Functions” on pages 226 to 232. Work on examples 2 and 3 on your own. After you finish, look at the solution in the reading.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Dirac Delta Function”**Link: Lamar University: Paul Dawkins’*Differential Equations*: “Laplace Transforms”: “Dirac Delta Function” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read pages 234 and 235. Work on example 2 on your own. After you finish, look at the solution in the reading.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Convolution Integrals”**Link: Lamar University: Paul Dawkins’*Differential Equations*: “Laplace Transforms”: “Convolution Integrals” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read pages 237 (start with example 2) and 238.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3B. Derivative Formulas; Solving ODE’s”**Link: MIT: Professor Arthur Mattuck’s*18.03 Notes and Exercises*: “Exercises”: “Laplace Transform”: “3B. Derivative Formulas; Solving ODE’s” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 3B-1, 3B-3, 3B-4, 3B-5 and 3B-6. When you finish, please check your answers to these exercises with “Section 3 Solutions.”

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3C. Discontinuous Functions”**Link: MIT: Professor Arthur Mattuck’s*18.03 Notes and Exercises*: “Exercises”: “Laplace Transform”: “3C. Discontinuous Functions” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 3C-1, 3C-2, 3C-3, 3C-4 and 3C-5. When you finish, please check your answers to these exercises with “Section 3 Solutions.”

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3D. Convolution and Delta Function”**Link: MIT: Professor Arthur Mattuck’s*18.03 Notes and Exercises*: “Exercises”: “Laplace Transform”: “3D. Convolution and Delta Function” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 3D-1, 3D-2, 3D-3, 3D-4, 3D-5, 3D-6, 3D-7, and 3D-8. When you finish, please check your answers to these exercises with “Section 3 Solutions.”

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