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MA221: Differential Equations

Unit 7: Nonlinear ODEs   This course has concentrated on the properties and solutions of linear ODEs, but we have also seen methods that apply to the occasional nonlinear ODE.  While the detailed study of solution methods for nonlinear ODEs lies beyond the scope of this course, it is worthwhile looking at a few examples.

Unit7 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Find the solution for the Riccati equation. - Describe how Nurgaliev’s law can be used to model population dynamics. - Find the solution for the Lotka-Volterra equation. - Describe how nonlinear ordinary differential equations can be used to model inverted pendulum dynamics. - Describe how nonlinear ordinary differential equations can be used to model double pendulum dynamics.

7.1 Riccati Equations   - Web Media: SOS Math: Mohamed Amine Khamsi’s “Ricatti Equations”

Link: SOS Math: Mohamed Amine Khamsi’s “[Ricatti
Equations](http://www.sosmath.com/diffeq/first/riccati/riccati.html)”
(HTML)  
    
 Instructions: Click on the link above.  Read the entire article. 
Try to work through the examples on your own before looking at the
solutions in the article.  
    
 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”: “First Order ODE’s”: “1B. Standard First Order Methods” Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1B. Standard First Order Methods” (PDF)
     
    Instructions: Please click on the PDF linked above and go to page
    1.  Work with all items in exercises 1B-10.  When you finish, please check your answers to this exercise with “Section 1 Solutions.
       
      Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

7.2 Nurgaliev’s Law   - Reading: Wikipedia’s “Nurgaliev’s Law” Link: Wikipedia’s “Nurgaliev’s Law” (HTML)
 
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7.3 Lotka-Volterra Equation   - Reading: Wikipedia’s “Lotka-Volterra Equation” Link: Wikipedia’s “Lotka Volterra Equation” (HTML)
 
Instructions: Click on the link above and read the entire article.
 
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7.4 Segway Dynamics – Inverted Pendulum   - Reading: Wikipedia’s “Inverted Pendulum” Link: Wikipedia’s “Inverted Pendulum” (HTML)
 
Instructions: Click on the link above and read the entire article.
 
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

7.5 Chaotic Dynamics – The Double Pendulum   - Reading: Wikipedia’s “Double Pendulum” Link: Wikipedia’s “Double Pendulum” (PDF)
 
Instructions: Click on the link above and read the entire article.
 
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).