Selwyn Hollis Differential Equations with Boundary Value Problems, Prentice Hall, 1st edition, c2002.
Selwyn Hollis A Mathematica Companion for Differential Equations, Prentice Hall, c2003.
Coursework: There will be a weekly problem set due at the beginning of class on Fridays. I expect you to write a complete solution to every assigned problem. If you get stuck on a problem, or if you are not confident in your solution, please confer with fellow students, a tutor, or come ask me in office hours. Late homework will be accepted with a 5% point penalty compounded daily. For example, an assignment that is 1 day late will yield 95% of its on-time value, 7 days late yields 70% of on-time value, and 14 days late yields 49% of on-time value. Missing homework will receive a score of zero.
Exams: There will be two midterms and a final. Midterm 2 will cover the material after midterm 1. The final exam will be cumulative with an emphasis on the material after midterm 2. Missing an exam will result in an exam score of zero being assigned. Grades: Course grades are based on problem sets and exams. The weight each component shall receive is as follows: 34% problem set average, 22% Midterm 1, 22% Midterm 2, 22% Final Exam. When the final exam score exceeds the lesser of the two midterm scores, the final exam score will replace the lesser of the two midterm scores. The following distribution will be used to assign course grades: 100-93% = A, 92-90% = A-, 89-86% = B+, 85-83% = B, 82-80% = B-, 79-76% = C+, 75-73% = C, 72-70% = C-, 69-66% = D+, 65-63% = D, 62-60% = D-, Below 60 % = F. Calculators: Calculators are always permitted. Tutoring: The Student Math Center is open every Sunday and Monday from 7-9pm in Galileo 201. Additional tutoring may be requested from the Academic Support Center in Sichel 105. Academic Honesty: The Saint Mary's policies regarding academic honesty detailed in the student handbook apply to this course. I encourage you to work with other students on coursework, but your write-ups should be in your own voice, and consist largely of your own work. Where your argument depends heavily on another's work, say so. Course Calendar: The course calendar details the schedule of coursework, exams and breaks.
Homework and Announcements
A comparison of solution of a square-wave drive under-damped mass-spring system using Laplace Transforms versus using Fourier Analysis. Here is a plot of partial sums of the Fourier series of the signal.
The Practice Final is worth 20 points of extra credit if submitted by the start of the Final. Assignment 13 due Fri May 12
7.5 #1, 2, 3, 7, 8, 16, 17
7.6 #1, 4, 5, 7, 27, 28 (See "The Green's Function Revisited" on p.256 and Example 7 on p.257)
11.3 Solve #1 and #6. Then, using the Fourier Series for each phi you found in #1 and #6, solve the DE y''+9y=9phi(t). Assignment 12 due Fri May 5
7.2 #1, 3, 5, 7, 9, 17, 19, 21, 23
7.3 #1, 4, 7, 27, 28, 29, 35, 36
6.5mc Resolve 7.3#35, 36 using the technique of 6.5mc Example 6.5.2 on p.139
7.4 #1, 6, 7
6.5mc Resolve 7.4#1, 6, 7 using the techiques of 6.5mc Example 6.5.3 on pp.140-141 Assignment 11 due Fri Apr 28
6.5 #1-6
6.6 #1,3,5,7-9,12
6.4mc Read the section titled Playing with Sound on p.134
6.4mc #7 Hints: The DE has a typo: 4.0y should be 4.0y' Also see Example 1.2.2 on p.29 for a reminder on how to use DSolve.
7.1 #1,2,5,6,7,33,35
6.5mc Resolve 7.1 #1,5,33 using the LaplaceTransform and InverseLaplaceTransfom commands as described on pp.136-138 and in Example 6.5.1
The Tacoma Narrows Bridge is a classic example of resonance for the underdamped driven harmonic oscillator.
A demo of the undamped, underdamped, critically damped, and overdamped spring-mass system. Assignment 10 due Fri Apr 21
6.3 #1-6
6.4 #1-3, 13-15, 20 Assignment 9 due Fri Apr 7
5.6 #10, 13
6.6mc Use the SeriesDSolve command on p.152 of the Mathematica Companion to resolve 5.6 #10(b), 13(b), 18, 21
6.1 #1, 2, 11, 12, 16, 17
6.2 #1-3 Practice Midterm 2 is worth 20 points of extra credit if submitted by Wed Apr 5. Assignment 8 due Fri Mar 31
By rest solution the text means a solution satisfying the initial conditions y(0)=0, y'(0)=0.
5.5 #1, 2, 3, 5, 6, 8, 9
5.6 #1, 3, 5, 18, 21 Assignment 7 due Fri Mar 24
5.2 #2, 3, 5, 6, 12, 13
6.1mc Use the TimeStatePlot command to verify that your answers in 5.2 #12, 13 are roughly correct.
5.3 #1, 2, 3, 5, 9, 10
5.4 #1, 11-13, 21
Time-state trajectories for the simple harmonic oscillator, the damped harmonic oscillator and the non-autonomous DE x''+tx=0. Assignment 6 due Fri Mar 17
4.6 #1, 3-6, 11-13
5.1mc Resolve 4.6 #11-13 using the PhasePlot command.
5.1 #5-8
A phase portrait of the coupled system dx/dt=x+y, dy/dt=1-x^2-y^2. Assignment 5 due Fri Mar 10
4.3 #1, 2, 6, 7, 9, 11
4.5 #4, 10
4.3mc Use the methods of 4.3mc to solve 4.5 #5 Plot of direction fields and solution curves for the autonomous DE y'=y^2-1 including the stable and unstable solutions y(t)=-1 and y(t)=1. 20 points Extra Credit due Wed Mar 8
Construct a deck of 52 DE flash cards. Each card has a DE on one side. The other side has the DE name (i.e. first order linear homogeneous, Bernoulli, separable, etc.) and the DE solution method (i.e. the name of the method and the relevant formula, method, or variable substitution). Get creative! Assignment 4 due Fri Mar 3
3.5 p.77 #4, 5
4.1 #1, 2, 5, 6
4.1mc Resolve 4.1#1, 2 using the SeriesDSolve command of 4.1mc
4.2mc Resolve 4.1#5, 6 using the Picardstep command of 4.2mc
4.2 #1,3 5-8, 13
5 points extra credit: p.83 #11
A comparison of the methods of Cauchy-Euler, Taylor, and Picard to find approximate solutions to the DE y'=(1+t)sin(t) Assignment 3 due Fri Feb 24
3.2 #1-4, 10-12
3.3 #1-3, 5-8, 17, 18, 20
3.4 #1-4, 12, 13, 15
An example of using Euler's method to plot solution curves to the DE y'=y with initial condition y(0)=1.
Applying separation of variables to the DE y'=(1+sin t)/(1-3y^2) results in the equation y-y^3=t-cos(t)+C. Here is an example of using the ContourPlot command to plot solutions to this equation for various values of C.
An example of plotting direction fields, solution curves and isoclines for the DE y'+3y=6t using the DEPlot command. Assignment 2 due Fri Feb 17
2.2.3 #1,5,8
2.3 #7,12
3.1#2-5,11,13
3.1mc Resolve 3.1#2,3 using the methods given in 3.1mc
3.2mc Resolve 3.1#4,5 using the DEPlot command.
You will need to download the Hollis Mathematica package DEGraphics from the Hollis website to complete 3.2mc.
1. Go to the Hollis website.
2. Click the Download! button.
3. Follow installation instructions 1-4 including the extra instruction at the bottom for Mathematica 4.x or 5.0
Now you should be able to enter the command <<Diffeqs`DEGraphics` into Mathematica follwed by DEPlot[t^3+y^3,{t,-2,2},{y,-2,2}]; to a get a plot of a direction field and some solution curves.
In the references below, 1.3 denotes chapter 1 section 3 of Differential Equations, and 1.1mc denotes chapter 1 section 1 of the Mathematica Companion. Mathematica Companion problems to be submitted as printout of Mathematica session. Consult Chapter 0 of the Mathematica Companion for help using Mathematica. Assignment 1 due Fri Feb 10
1.3 #1-5,10-12,15-17,19,25,35,39,43,48
1.1mc #1(a)(e), 2(a)(e), 3(a)(e)
1.2mc #1(a),2(a)
2.1 Use (a) integrating factors, (b) variation of parameters to solve #1,3,5
2.1 #24,26
1.3mc Resolve 2.1#26 by plotting solution families
1.4mc Resolve 2.1#1,3,5 using the Integrate command.
1.5mc Resolve 2.1#24(b) using the NDSolve command.
