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.)
Reading: Sections 1.1, 1.2, 2.2. In section 2.4, also read Theorem 2.4.2 on p70, and the section on Implicit Solutions on p74. In section 2.8, read p112-113.
Section 1.1: 1, 6, 8, 10, 26, 32
Section 1.2: 3, 7, 8
Section 2.2: 2, 3, 7, 9, 12
Section 2.4: 7, 11, 13, 15, 16
Here are some online resources about isoclines. Most of them I stole form the ODE class at MIT, but don't let that scare you - they're very well explained and well made. 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.
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.
Reading: Sections 2.1, 2.4, and 2.5. Also USE THE APPLET mentioned above to explore isoclines.
Section 2.1: 5c, 7c, 14, 19, 22, 23, 28
Section 2.4: 2, 3, 4
Section 2.5: 2, 4, 8, 10, 13
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?
Reading: Sections 1.3, 2.7, 7.1
Section 1.3: 1, 3, 4, 17, 18, 20
Section 2.4: 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 .
Ben's Problem 3 (Integration tricks)
a) I know that the antiderivative of (3t2 - 4t + 5) e2t is of the form (at2 + b t + c) e2t + C for some numbers a, b, c (where C is the extra constant of integration). Find these numbers by differentiating, not by integrating!
b) I know that the integral of 2 t * sin t is of the form (at + b) sin t + (ct + d) cos t + C for some numbers a, b, c, d . Find these numbers by differentiating, not by integrating!
I have posted the Practice Midterm. Solutions are posted here.
The practice midterm is longer than the actual midterm. I wanted to include a variety of problems.
We will spend Wednesday on review.
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
We've talked in class about going back and forth between different ways of writing sinusoidal functions. This is not in the book. It is explained well in the first 3 pages of these lecture notes. Here is an applet where you can compare the different forms. The goal in the applet is to try to make the red and green waves overlap.
Reading: Sections 3.3, 3.4, 4.2, ignoring anything about the Wronskian. The first 3 pages of 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.
Reading: Sections 3.5, 4.3.
Section 3.5: 2, 4, 5, 11, 14, 17
Section 4.3: 3, 4, 8, 11, 14, 15, 16, 18
Here is a resource about the exponential and sinusoidal response formulas.
I have posted the practice midterm 2. Solutions are here.
Reading: This handout, Sections 3.7, 3.8
Section 4.3: 1, 2, 6, 7, 13, 17
Ben's problems: in this pdf. Solutions are posted here.
Reading: Sections 7.1, 7.2, 7.3 (focus on eigenvectors and eigenvalues)
Section 7.1: 2, 4, 5 (This is recalling something discussed much earlier in the class, and is described on p361)
Section 7.2: 1, 2, 22, 23, 24. Also compute the determinants of the matrices in problems 10, 11, 12, 13.
Section 7.3: 16, 17, 18, 20.
Here are some applets that draw phase plane diagrams. They are really very nice!!
Here is an applet that lets you input the bottom row of a matrix whose top row is [0,1]. It then draws the phase plane for you. If you prefer, instead of entering the bottom row, you can just enter the trace and the determinant! It will draw any solution you click on as well.
Here is a weirder applet that can draw the phase plane diagram of any matrix, but its not so easy to get the matrix you want. Compute the trace and determinant first, and then screw around.
Reading: Skim section 7.4, read 7.5 and 7.6. There is also a nice summary for studying later in 9.1, with an important chart on p507.
Section 7.5: 2, 3, 4, 5, 8, 15, 16, 26, 27
Section 7.6: 1, 2, 3, 5, 10
Here is a study guide that I've written for the whole quarter.
A quasi-practice-final has been posted. Although the true final will be cumulative, this focuses only on the material since the second midterm. Solutions are here. Some phase portraits in the solutions are here.