# MA221: Differential Equations

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.

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.

• 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.

• 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.”

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.

• 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.

• 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.

• 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.

• 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.”

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.

• 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.

• 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.

• 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.

• 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.

• 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.”