# MA221: Differential Equations

Unit 4: Systems of Linear Differential Equations   In the previous units, we worked with differential equations with one unknown function.  However, some problems involve several functions that depend on a single variable.  In these cases, a system of ordinary differential equations can be created to model the problem.

The concept of the differential operator will help us develop techniques for solving systems of linear ODEs.  Briefly, if we define the differentiation operator Dx = d/dx, so that DxY = dY/dx, a general linear ordinary differential equation can be written as a polynomial in Dx.  This allows us to form an equivalence between a system of linear ODEs and a matrix equation, which can then be solved as an eigenvalue problem using linear algebra.  This is quite a boon, as most problems in linear algebra can indeed be solved!

Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Find the solution for a system of ordinary differential equations. - Interpret the role of each differential equation within a system of ordinary differential equations in modeling a selected application. - Create a system of ordinary differential equations to model a selected application. - Find the solution for a system of ordinary differential equations within applications involving spring-mass dynamics. - Describe all components of the differential equation used within the predator-prey model. - Describe all components of the differential equation used to describe the double pendulum. - Distinguish between the usage of existence and uniqueness theorems to support claims which arise within selected applications. - Convert an nth order ordinary differential equation into an n-dimensional system of first order ordinary differential equations. - Apply selected linear algebra-based methods to find the solution for the systems of linear ordinary differential equations.

4.1 Examples of Systems of ODEs   4.1.1 Mass and Spring Systems   - Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Systems of ODEs”: “3.1 Introduction to Systems of ODEs” Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Systems of ODEs”: “3.1 Introduction to Systems of ODEs” (PDF)

Also available in:
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``````[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/08/MA221-book-Notes-on-Diffy-Qs-Jiri-Lebl.epub)

Instructions: Click on the link above and then click on the link
section titled “3.1 Introduction to Systems of ODEs” on pages
85-88.

It is attributed to Jiri Lebl, and the original version can be
found [here](http://www.jirka.org/diffyqs/) (PDF).

``````

Also available in:

``````[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/08/MA221-4.1.2-Frank-Hoppensteadts.epub)

article.

kind permission of Frank Hoppensteadts, and can be viewed in its
original form
[here](http://www.scholarpedia.org/article/Predator-prey_model).
reproduced in any capacity without explicit permission from the
``````

4.1.3 The Double Pendulum   - Reading: The MathWorks: Cleve Moler’s Numerical Computing with MATLAB: “Ordinary Differential Equations” Link: The MathWorks: Cleve Moler’s Numerical Computing with MATLAB: “Ordinary Differential Equations” (PDF)

Instructions: Click on the link above.  Click on the link “Ordinary Differential Equations (53 pages).”  Go to page 49 and read the item numbered “7.23.”  Read only up to page 50.

4.2 Existence and Uniqueness Theorems   - Reading: Michigan State University: Sheldon Newhouse’s Math 235 Lecture Notes: “16. Systems of Differential Equations” Link: Michigan State University: Sheldon Newhouse’s Math 235 Lecture Notes: “16. Systems of Differential Equations” (PDF)

Instructions: Click on the link above, scroll down, and, under the title “Lecture Notes,” click on the link for “16. Systems of Differential Equations.”  Go to page 5.  Read the section titled “2 Systems of Differential Equations” on pages 5-11.

• Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Systems of ODEs”: “3.3 Linear Systems of ODEs” Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Systems of ODEs”: “3.3 Linear Systems of ODEs” (PDF)

Also available in:
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EPUB

Instructions: Click on the link above and then click on the link that says “Download the book as PDF.”  Go to page 99.   Please read the section titled “3.3 Linear Systems of ODEs” on pages 99-102.

Terms of Use: The work above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML).  It is attributed to Jiri Lebl, and the original version can be found here (PDF).

4.3 Conversion to Systems of First-Order ODEs   - Reading: Wikipedia’s Ordinary Differential Equation: “Reduction to a First Order System” Link: Wikipedia’s Ordinary Differential Equation: “Reduction to a First Order System” (PDF)

Instructions: Click on the link above.  Read the sections titled: “Reduction to a First Order System” and “Linear Ordinary Differential Equations.”

Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML).  It is attributed to Wikipedia, and the original version can be found here (HTML).

• Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4B. General Systems; Elimination; Using Matrices” Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4B. General Systems; Elimination; Using Matrices” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 4B-1 and 4B-2 on page 1.   When you finish, please check your answers to these exercises with “Linear Systems Solutions.”

4.4 Linear Algebra-Based Solutions of Systems of Linear ODEs   - Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Systems of ODEs”: “3.4 Eingenvalue Method” Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Systems of ODEs”: “3.4 Eingenvalue Method” (PDF)

Also available in:
HTML

``````[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/08/MA221-book-Notes-on-Diffy-Qs-Jiri-Lebl.epub)

Instructions: Click on the link above and then click on the link
the section titled “3.4 Eingenvalue Method” on pages 103-109.

It is attributed to Jiri Lebl, and the original version can be
found [here](http://www.jirka.org/diffyqs/) (PDF).
``````
• Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4B. General Systems; Elimination; Using Matrices” Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4B. General Systems; Elimination; Using Matrices” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 4B-4, 4B-5, and 4B-6 on pages 1 and 2.   When you finish, please check your answers for these exercises with “Linear Systems Solutions.”

• Assessment: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4C. Eingenvalues and Eigenvectors” Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4C. Eingenvalues and Eigenvectors” (PDF)

Instructions: Click on the PDF linked above and go to page 2.  Work with exercises 4C-1 and 4C-5.   When you finish, please check your answers to these exercises with “Linear Systems Solutions.”

• Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4D. Complex and Repeated Eingenvalues” Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4D. Complex and Repeated Eingenvalues” (PDF)

Instructions: Please click on the PDF linked above, and go to page

1.  Work with exercises 4D-1, 4D-2, and 4D-3.   When you finish, please check your answers to these exercises with “Linear Systems Solutions.”