# MAT 256, Winter 2015

Welcome to Math 256, Introduction to Differential Equations. The syllabus is here. It contains information on class times, exam times, office hours, contact info, etcetera.
This website will be used to make announcements, post homework assignments, post practice exams and solutions, etcetera.

Office hours and midterm times have been agreed upon, and are listed in the syllabus.

HW is due at the BEGINNING OF CLASS each Wednesday. (You may also put it in my mailbox in Fenton hall if you don't come to class.)

HW 1 (due W 1/14) - a little longer than the average homework, but best to start strong, eh?

Reading: Sections 1.1, 1.2, 2.2, and 2.5
Section 1.1: 1, 6, 8, 10, 26, 32
Section 1.2: 3, 7, 8
Section 2.2: 2, 3, 7, 9, 12, 24
Section 2.5: 2, 4, 8, 13

Here are some online resources about isoclines. Most of them I stole from the ODE class at MIT, but don't let that scare you - they're accessible and well done, mostly. You can find many more by googling the words isocline and funnel.
An applet where you can select from a list of differential equations. If you click on the graph it will draw the solution through that point. If you move the slider, it will draw an isocline. I highly recommend playing around here! The demo video is also very instructive! It discusses fences and funnels around 5:45 and following.
A direction field viewer for more practice with other equations. It won't draw isoclines though. This is a downloadable java app.
In this lecture is a gentle introduction to isoclines, somewhere around 35 minutes. I point this out to indicate that one can find many excellent lectures online, if one is interested.
A not-wonderfully-written discussion of fences, funnels, and separatrices.
A first introduction to isoclines here.
A nice set of introductory practice problems here. Solutions are here.

HW 2 (due W 1/21) - two isocline problems not in your book below

Section 2.1: 5c, 7c, 14, 19, 22, 23, 28
Section 2.4: 2, 3, 7, 10, 14

Ben's Problem 1 Consider the differential equation y' = x + y.
a) On a graph ranging from values -3 to +3, draw the isoclines for slopes -2, -1, 0, +1, and +2.
b) Find a linear solution, and prove that it is the only linear solution.
c) Is the linear solution in a funnel or an anti-funnel? Is it a separatrix? Specify two isoclines to make your point.
d) Consider the solution with y(0) = 1 . How many maxima and minima does it have? Give an argument for why.
e) Consider the solution with y(0) = -2 . How many maxima and minima does it have? Give an argument for why.

Ben's Problem 2 Consider the differential equation y' = y + 2x - x2.
a) On a graph ranging from values -3 to +3, carefully draw the isoclines for slopes -2, -1, 0, +1, and +2.
b) Sketch the solutions with initial values y(0) = -1, 0, +1.
c) Are there any funnels or antifunnels? Why or why not?
d) Note that y=x2 is a solution. Find the general solution to this 1LODE. How good were your earlier sketches?
e) Is the solution y = x2 a separatrix?

HW 3 (due W 1/28)

Section 1.3: 1, 3, 4, 17, 18, 20
Section 2.4: 22, 24, 25
Section 2.7: 1(abd), 2(abd) - use a calculator

Ben's Problem 1 Consider the first order system satisfying y' = x + y and x' = ty+x+1, with initial value y(1) = 1 and x(1) = 2. Use the Euler method with step size h = .5 to approximate the values of y(2.5) and x(2.5).

Ben's Problem 2 For the two differential equations below, find the general solution. Is there a separatrix? Why or why not? If there is a separatrix, find a number b such that, whenever y(1) is greater than b the solution goes to infinity, while whenever y(1) is less than b the solution goes to negative infinity.
a) y' - 2y = et .
b) y' - y = e2t .

I have posted the Practice Midterm 1. We will discuss it in class on Friday, so please try it before then. Solutions are posted here.
This practice midterm is LONGER than the midterm will be, but I wanted to include a larger variety of questions. Give yourself 70 minutes.

HW 4 (due W 2/4)

Reading: Sections 3.1, 4.1 first two pages, 4.2 ignoring complex roots
Section 3.1: 2, 4, 6, 8, 9, 12, 18, 20
Section 4.1: 19(ab)
Section 4.2: 11, 14, 16, 17, 24

Here are some resources about sinusoidal functions. Some lecture notes, and an applet. The goal in the applet is to try to make the red and green waves overlap.

HW 5 (due W 2/11)

Reading: Sections 3.3, 3.4, 4.2, ignoring anything about the Wronskian for now. Section 4.1 from these lecture notes.
Section 3.3: 2, 4, 8, 10, 19, 21, 29, 32
Section 4.2: 1, 3, 13, 18, 20, 22

Ben's Problem This problem is about writing sinusoidal functions in different forms. This is explained in the lecture notes above.
a) Write 3 cos 2t + 4 sin 2t in the form A cos(wt - s) , for some A, w, s.
b) Write -2 cos 6t - sin 6t in the form A cos(wt - s) , for some A, w, s.
c) Write 2 cos(5t - pi/4) in the form a cos wt + b sin wt , for some a, b, w.
d) Write 4 cos(t - 4 pi/3) in the form a cos wt + b sin wt , for some a, b, w.

HW 6 (due W 2/18)

Section 3.5: 2, 4, 5, 11, 14, 17
Section 4.3: 3, 4, 8, 11, 14, 15, 16, 18

I have posted the practice midterm 2. Solutions are here.

Here is a resource about the exponential and sinusoidal response formulas.

HW 7 (due W 2/25)

Reading: This handout, Sections 3.7, 3.8
Section 4.3: 1, 2, 6, 7, 13, 17
Ben's problems: in this pdf. (NOTE: I made an error defining the gain in a previous version of this pdf. It is now corrected.) The solutions are here.

Some more resources for you. This is an excellent free online textbook on ODEs. Chapter 2.6 is a good read for this week.
There are also the MIT notes which I've been calling upon. The apples in L16 are definitely worth playing with!

HW 8 (due W 3/4)

Reading: Section 7.1, Section 7.2 but ignore "multiplication of vectors," Section 7.3 until the very top of p84
Section 7.1: 3,5
Section 7.2: 1,2,11, 13, 21, 23 ,24
Section 7.3: 8, 10, 11, 13, 14, 15

HW 9 (due W 3/11)

Reading: Section 7.3 eigenvectors and eigenvalues, Section 7.5, Section 7.6
Section 7.3: 17, 18, 20, 23
Section 7.5: 2, 3, 4, 8, 15, 29
Section 7.6: 2, 3, 4, 5, 10

Suggested reading for final week: Section 7.8 to the end of p433 (not on the final), Section 9.1. Note the figure on p507.

I wrote a study guide. Hopefully it comes in handy.

I have posted a quasi practice final. The final is comprehensive, but the quasi-practice final gives only questions for the last third of the class (things which haven't been covered in previous midterms. Solutions are here. The phase portraits in the solutions are here.

Ben Elias
Department of Mathematics
Fenton Hall, Room 210
University of Oregon
Eugene, OR 97403
Phone: (541) 346-5629
Fax: (541) 346-0987
e-mail: bezzzzlizzzzas@uorezzzzgon.edu