## Ordinary Differential Equations Fall 2014

### Instructor: Bard Ermentrout

#### Office Hours: MW 2:00-3:00 PM Office: Thackeray 502 Phone: 624-8324 EMAIL: bard+@pitt.edu WWW: http://www.pitt.edu/~phase

All homework is due the week following the assignment. Grades will be based on:
• Written homework 40%
• 2 Exams 20% (Midterm) Final 40%%

The book is Elementary Differential Equations and Boundary Value Problems The authors are Boyce and DiPrima

Please note This is the tenth edition! (It seems to differ little from earlier editions, so you are probably safe with them. I have the 9th as well so I can help you align the exercises)

### Software

Some of the exercises will require you to use a computer to create pictures. There are several ways to do this. Some of you are probably familiar with MatLab which has something called PPLANE which will be helpful. I wrote a software package XPPAUT for solving and graphing differential equations. This runs on all PCs and also runs on iOS devices (sorry, no Android). You can get this at This site . I can help you get it on your computer as it requires a small amount of effort

### Syllabus:

• Week 1: (8/25-8/29) chapter 1.1;2.1;2.2

Homework due: 9/5: 1.1:15-20,23,24,26,29,30;2.1:13,15,16,31,32;2.2:1,5,8,9,10,17,31,37,36;

• Here is a handout for using XPP and doing some of the computer problems How to plot
• Simple XPP code for direction fields
• Run this in XPP. Click on (D)ir.field (S)caled and then Return to accept the default. See the nice direction fields!
• Click (I)nitialconds m(I)ce and click around on the screen near the dashed line. See the trajectories. Tap ESC when done.
• Take a screen shot of this to print it if you want
• Week 2 -- 9/3,9/5 2.3,2.4. HW 2.3:2,9,19,29,32;2.4:1,3,5,7,14,15,23,27,28; DUE 9/12
• Week 3 -- 9/8-9/12 2.5,2.6,2.8 HW 2.5: 1,3,4,8,11,13,20,25; 2.6:1,3,5,9,13,15,18,19,25; 2.8: 5a-c DUE 9/19
• BONUS ROCKET PROBLEM and NUMERICAL Exercise Due 9/26
• Week 4 -- 9/15-9/19 3.1-3.3
Homework Due 9/26
• 3.1:1,7,9,12,17,20,23,28
• 3.2:1,2,4,7,12,13,16,17,23,28,29
• 3.3:1,4,6,10,15,21,34,35,39
• Week 5 -- 9/22-9-26
Homework Due 10/3
• 3.4:7,11,12,17,20,21
• 3.5: 1,6(4),10(8),14(12),16(14),20(18) [Note 9th edition in parentheses]
• 3.6: 1,2,9,13
• Week 6 -- 9/29-10/3 (Due 10/10)
• 3.7: 1,5,7,13,18
• 3.8: 1,11,17,24
• 4.1:3,6,7,11,15,24 (you can assume without proving it, the result of problem 20 on page 225)
• Week 7 - 10/6-10/10
• Week of October 13-17 (Recall we have class Tues/Wed/Fri this week )
• HOMEWORK DUE 10/24:
• Chapter 4. 4.2:11,18,21;4.3:1(see 4.2,11);4.4:1
• Chapt 7.1 1,4,6,23
• 7.3:1,4,15,18,21,23
• Week of October 20-24 (HOMEWORK DUE OCT 31 )
• 7.4:2abc,6
• 7.5:1,2,5,7,11,15,16,20,24,25,27,31
• 7.6:1,3,5,13,14,28
• Week of Oct 27-31 (Homework DUE Nov 7 )
• 7.7:1,3,5
• 7.8:1,2,11
• Let A be a 3x3 matrix with eigenvalues -1, -1+2 i. Express exp(At) in terms of the matrix A. (Use Fulmer's method)
• Use Fulmer's method to do problem 1 in 7.7
• Week of Nov 3-Nov 7 (Homework due Nov 14)
• 7.9:3,12
• 9.1: 1,3,5,6,13
• 9.2:1,4,5,9,17,21
• Bonus Problem (due last week of class)
• At 12 noon, a snowplow begins plowing the street. He plows 2 miles the first hour and 1 mile the second. What time did it start to snow?
• The Lotka-Volterra equations satisfy

dx/dt = x *(a-b*y)

dy/dt = y*(-c+d*x)

They have infinitely many periodic solutions. Let T be the period of one of the solutions and let x(t),y(t) be the solution. Compute the average values of x(t),y(t):

xbar = (1/T) integral (x(t),t=0..T)

ybar = (1/T) integral (y(t),t=0..T)

• Last week of class - chapter 9.7
• Review and sample Final
• FINAL EXAM THURSDAY December 11 2:00-3:50 PM - CL 230
• The exam will be OPEN BOOK/OPEN NOTES
• Sample Final Exam
• The exam will cover the following topics:
• Higher order linear differential equations
• Routh-Hurwitz stability
• method of undetermined coefficients
• variation of parameters
• Systems of linear constant coefficient equations
• Eigenvector/eigenvalue methods
• Matrix exponential
• Fulmers method
• Variation of parameters
• Two-dimensional phase plane
• Phase portrait
• Trace-determinant plane
• Nullclines
• Stable/unstable eigenvectors
• Nonlinear planar systems
• Phase portraits
• Integrable systems (exact ODES)
• Nullclines
• Singular points/ linearization/stability
• Multi-species problems (predator/prey and competition)
• Bifurcations (change in behavior as parameters change)