fAMAT311  Ordinary Differential Equations

MWF 9:20-10:15  ES 147


Instructor: Professor Edward Thomas  ES 132F

Phone 442-4623

Best way to reach me is by email: et392@albany.edu

Office hours: MWF 10:25-10:45, 12:50-1:30, 2:45-3:15 and by appointment

A link to the class webpage will be found at : www.albany.edu/~et392/AMAT311.htm


Text: Elementary Differential Equations by Edward and Penney; we will be covering Chapters 1 through 4, plus other topics if time permits.


   You need to have this text on the first day of class, not at some indeterminate later date (see the next paragraph.) If you show up on the first day of class without the text, I will take this as a sign of LACK OF PREPARATION.


  In this course, you can learn both technique and theory by doing problems. So I am going to assign problems every single day, starting on day one. They will be collected, graded and returned to you at the next meeting and will serve as the springboard for what comes next. You should assign them high priority…I’m not kidding on this.


  Daily assignments will count one third of the grade. The other two thirds will come from a Midterm and a Final.



   Let’s articulate  some ground rules:


> First...there will be ABSOLUTELY NO CELL PHONES, LAPTOPS or any other type of electronic devices in use during class. Please take care of business and TURN THEM OFF before you enter the classroom.


>Second…please DO NOT come to class late as it is disruptive. Be in your seat, mentally alert and ready to participate, at 9:20 when class begins.


> Third  If you get sick or have some other kind of emergency, please get in touch with me as soon as you can so we can work things out.


 > Fourth, classes begin on Monday, August 25th. ( You wouldn’t believe it but in the past some peeps thought they could begin classes on a day of their own choosing. That was a BIG mistake.)



1)   page 17 # 1-6 and 8,9  ( Geez! Already?  Ashley has pointed out that there is a mistake in the answer section for this assignment !)

2) page 18 # 24, 26 and 36 plus page 43 # 1-4  Some remarks:   On problem 26, tMAX is approximately20.4 seconds.

On problem 36, you are given that xMAX = 2.25, from which you want to compute v0. First compute tMAX. You’ll find that tMAX =v0 / gE ( Earth’s gravitational constant) Then using that, you find that xMAX =v02 /2gE. and from that you can figure out  v0.

3) page 43 # 21, 22, 25

 4) Using the value of k that we found in class, predict the U.S. population in 1920 ( compare with the actual value of about 106 million) Plus, on page 43, do numbers 33-38

 5) page 84 #32 ( just do the derivation as discussed in class…i.e., go from equation (*) to this form of the solution) and # 29 ( on part c , maybe just see how accurately the Verhulst model predicts P(2000)or P(1990)

 6) page 44-45 # 43, 48, 65

7) page 54 # 2, 3, 4, 8, 15, 17, 22, and 24

8) page 55 # 33, 35

 9) A handout on exactness. I’ll post a couple of extra copies on my door.

10) pg 73 # 33, 35, 37, 39 Do these by the systematic way introduced in class, please.

11) Using the numbers provided in class ( copies on my door) compute escape velocity for the Sun, Earth, Moon and Antares. Then, for the Sun, Moon and Antares,  compute what radius will produce a black hole…as we did in class for the Earth.

12) pg 111 # 1, 2, 5, 6

13) A handout with 8 problems on solving second order equations..copies on my door ( The equation in problem 4 should end in 25y, not just 25. Sorry for the misprint.) Comments added Saturday morning….if you have complex roots a+bi and a-bi and the a is equal to 0, the corresponding real solutions are just cos(bx) and sin(bx), since eax=1. Also, in the first problem, the roots are real; they are approximately -3.62 and -1.38.

14) Solve the spring/mass equations with the following data: (1) m =4  k= 16  x(0)= 2  x’(0) = -3 and (2) m = 1  k= 9  x(0) = -1  x’(0) =0

15) page145 #1-4


    Announcement about late homework….I’m afraid I have been encouraging bad habits among a very few of us. From now on, unless you have spoken with me in advance, as soon as I start grading an assignment any previous assignments that have not been handed in will be regarded as late. They may incur a late penalty or, if they’re REALLY late, they may not get graded at all.


16) page 147 # 13 a, # 15,16  In each case simply find the solution  using initial data, and, in #13, find the time to reach max displacement: tmax =?

17) page 147 # 18 and 20

18) page 161 # 1, 2, 3

 I also passed out a list of topics from which exam questions will be selected. I’ll post copies on my door.


  MIDTERM  Monday  October 13 th

19) page 161 # 31, 32, 3320)  Calculate the response amplitude for the system x” + 16x = sin ( omega)t when omega = 4.1, 4.08, 4.05 , respectively. Suppose the system will collapse if the response amplitude reaches 5 units. What omega will produce this?   Then, let’s do an experiment. Suppose we look at x” + 9x = cos 3t. Show that the method of undetermined coefficients fails if we try a trial solution of the form xT = A cos 3t. ( We’ll see what to do about this next time.)

20) #1) for the equation x” + 9x = cos3t, substitute the trial solution xT = t (A cos3t + Bsin3t) and, after much computation, verify that the particular solution is            xp =(1/6) ( t sin3t)  #2) Then, using the initial data x(0) =1 and x’(0) = 5, find the complementary solution xc = sqrt (34)/3 cos ( 3t – phi).

The point here is to illustrate the dominance of xp when you have pure resonance.  #3) This problem gets us started on resonance in damped systems. Take the equation mx” + cx’ +kx = F0cos (omega)t. We want to work out the general form of the steady state solution, xp, so substitute the trial solution, xt = A cos ( omega)t + B sin(omega)t and calculate what A and B are in terms of K,m,omega, c and F0. We’ll go from there next time.

21) Starting with the values of A and B that we found in the last problem, show that the response amplitude, denoted C( omega), is equal to F0 divided by the square root of ( k – m(omega)2)2 + (c(omega))2.  We want to rewrite this in terms of the RLC constants. Show that when we use current I as the dependent variable,                C(omega) becomes E0 divided by the square root of ( 1/(omega)C –(omega)L)2 +R2.

    And, from there, we saw in class how to tune a radio….

22) page 206 # 2, 3, 4 plus complete the solution of the equation: y” + 9y =0

23) No assignment

24) 1) Find the  series expansion for the Bessel function of order 1, J1 (x), as we did for J0 (0) in class. Don’t forget to multiply by a factor of ½.

       (2) Verify that the derivative of x J1(x) = xJ0(x) by writing out 5 or 6 terms of each side.

25) a) Compute the solution of  Bessel’s equation of order 2 by our techniques, call it y(x). The Bessel function of order 2, J2(x) is a multiple of this y(x), i.e.,       J2(x) = K times y(x). What is K?  b) Compute the Laplace transform of sin (t) from the definition as an indefinite integral. You can certainly look up the formula for the required antiderivative, but then you need to take a limit.

26) With a minimum of work, find the formula for the Laplace transform of the second derivative of f(t), as discussed in class.

27) page 287 # 2, 3, 4,5

28) page 314 # 1, 2, 3, 5, 6

29) page 314 # 11, 12, 13 ( plot the graph, find the formula  for f(t), and then transform)  and page 315 # 31 ( use the stepping stones given in class)

30) page 324 #2 plus:  solve  x” + 9x = 4 times (delta ^ 6 Pi) with x(0) =1, x’(0) = 0

31) page 82 # 5 and 7 plus: solve dP/dt = kP(P-200) with k = .003. Evaluate the constant of integration, c , in term of P0 and thus solve for P(t) in terms of P0.

32) Take the in-class doomsday/ extinction problem stated above, with P measured in thousands and t measured in DECADES ( not years). Figure out when doomsday will occur if P0 = 250. Then P0 = 300.  What initial population will give tD = 25 years?

33) pages 82,83 do the following problems with the indicated numerical changes : # 9, P0 = 160 and initial growth rate 80 ( round off fractions to nearest rabbit),

# 13 …on part b, P0 = 12 and P(18) = 20. # 14 as is, # 28, dx/dt = .0005 x2 - .02 x  (a) P0 = 25, (b) P0 = 50 ( round off to nearest gator )


) No assignment ..Happy Thanksgiving

34) A problem on solving for I as a function of S in the SIR model of epidemics.



Final Exam options: Wednesday Dec. 10th either 9:15 -10:45 or 12:15 – 1:45

                                      Friday Dec 12th   9:30-11

                                      Thursday Dec 18th  9:00-10:30