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.)

Assignments:

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, t_{MAX}
is approximately20.4 seconds.

On
problem 36, you are given that x_{MAX}_{ }=
2.25, from which you want to compute v_{0. }First compute t_{MAX}_{. }You’ll find that t_{MAX}_{ =}v_{0} / g_{E}_{ }( Earth’s
gravitational constant) Then using that, you find that x_{MAX}_{
}=v_{0}^{2 }/2g_{E. }and
from that you can figure out v_{0. }

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 e^{ax}=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: t_{max}
=?

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 x_{T}_{
= }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 x_{T} = t (A cos3t + Bsin3t) and, after much
computation, verify that the particular solution is x_{p}
=(1/6) ( t sin3t) #2) Then, using the
initial data x(0) =1 and x’(0) = 5, find the complementary solution x_{c}
= sqrt (34)/3 cos ( 3t – phi).

The
point here is to illustrate the dominance of x_{p}
when you have pure resonance. #3) This problem gets us started on resonance in damped systems.
Take the equation mx” + cx’ +kx = F_{0}cos
(omega)t. We want to work out the general form of the
steady state solution, x_{p}, so substitute
the trial solution, x_{t}_{ }= A cos
( omega)t + B_{ }sin(omega)t and calculate what A and B are in terms of
K,m,omega, c and F_{0}. 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 F_{0}
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 E_{0 }divided by the square root of
( 1/(omega)C –(omega)L)^{2 }+R^{2}.

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, J_{1} (x), as we did for
J_{0 }(0) in class. Don’t forget to multiply by a factor of ½.

(2) Verify that the derivative of x J_{1}(x)
= xJ_{0}(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, J_{2}(x) is a
multiple of this y(x), i.e., J_{2}(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 P_{0}
and thus solve for P(t) in terms of P_{0.}

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 P_{0 }= 250. Then P_{0}
= 300. What initial population will give
t_{D} = 25 years?

33)
pages 82,83 do the following problems with the
indicated numerical changes : # 9, P_{0} = 160 and initial growth rate
80 ( round off fractions to nearest rabbit),

#
13 …on part b, P_{0} = 12 and P(18) = 20. # 14
as is, # 28, dx/dt = .0005 x^{2 }- .02 x (a) P_{0 = }25,
(b) P_{0} = 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. 10^{th}
either 9:15 -10:45 or 12:15 – 1:45

Friday
Dec 12^{th} 9:30-11

Thursday
Dec 18^{th}
9:00-10:30