AMAT311 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 techniques
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 Wednesday
January 22^{nd}. ( 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 # 1,2,3,6,8,9,and 11,12

2) First, in the problem we analyzed in class, with
initial velocity 50 and initial height 100, what is the impact velocity, v_{THUD}_{
}? Next,
on page 18, let’s do # 24,26 and 31.

3) Page 43 # 4,13, 21, 22,25

4) Page 43 # 33-37

5) Both problems have to do with the logistic equation
which we began discussing in class. First, expand 1/P(150-P)
by partial fractions and integrate. Second, after integrating, show that P/(150 – P) = C e ^{(.06)t} We’ll pick it up there next time.

6) Using the
values of k and M postulated in the Verhulst model,
predict the US population values for the years 1845 and 1895. ( These are remarkably accurate.) Next, in the Doomsday/Extinction model
which we started in class, show that if P_{0} = 200 then Doomsday will
occur sometime in the 23^{rd} year.

7) Page 54 # 1-4,
17, 18, 24. Put each of these in linear form and calculate the integrating
factor. You can stop there. Make two copies…one to hand in and one to keep/

8) Page 54 Finish #1-4, 18 and 24

9) A handout sheet on a linear equation which arises in
connection with a falling body encountering air resistance. YOU KINDA HAD TO BE
THERE.

10)
A handout on
exact equations

11)
page98 # 2 -12 . As discussed in class, just test each equation to
determine if it is separable, linear or exact (or none of the above) and stop
there…you don’t need to solve.

The Midterm
Exam will be on Monday, March 10^{th}

12) page 112 # 33-38

13) On page
134, take problems 3 – 9. Figure out which of them has complex roots and, for
these, write out the general solution.

14) page 145 # 1 and 2

15) page 145 # 3 and 4

16) page 147 # 14 part (a)

17) page 161 # 1, 2, 3

18) page 161 # 31, 32, 33

19) A handout
sheet with a problem on forced oscillations and resonance.

20) page 171 # 11, 12 ( omit finding c_{1 }and c_{2 })

21) First verify formula (21) on page 168 for
the equations in problems 11 and 12 on page 171 (which we just did for
homework.) ( Your answers for C should be sqrt(10)/4 and 5/3 sqrt(29), resp.) Then, do # 17 and 18 on page 171 except
don’t bother to graph C as a function of omega

22) A handout
asking you to take the formula for response amplitude in a mechanical system
and convert it to the corresponding formula for an electrical system. I will
leave some copies of the handout on my door. Added on Tuesday morning….
it seemed to work
out better for me if I took equation (2) , multiplied top and bottom by omega
(w), then took the w in the denominator into the square root sign, did some
algebra and THEN replaced L by m, R by c, 1/C by k and wE_{0} by F_{0
}to get the form (1)_{.}

23) Compute the
Laplace transforms of : sin(5t),
e^{3t}, and cos(t)…do these as improper integrals with limits
and everything as shown in class. ( Who knows? This
may be our last chance to do things this way.)

24)(a) Compute
L(t) as an improper integral. (b) Finish off the computation of L(u(t-a)) (c) Let
g(t) = 1 – u(t-a). Compute L(g(t)) using known
transforms and linearity. (d) Let h(t) = 12[ u(t-10) –
u(t-30)]. Draw the graph of h(t) and tell what it
might represent physically. Then, using linearity and known transforms, compute
L ( h(t))

25) Solve the following using Laplace
transforms: (1) x’ -4x = 1 , x(0) arbitrary ; (2) x’ + x = e^(2t) , x(0)
arbitrary; and (3) x’ -5x = sint(t) , x(0)
arbitrary

26)
page 287 # 1, 2, and 9

27)
a) Derive the formula for the Laplace transform of a
second derivative, using the formula for the first derivative. b) Following the
idea in class, derive the formula for L(t^{3}),
L(t^{4}) and then extrapolate to
get L(t^{n})

28)
page 314 # 1, 2, 3 , 6, 7, 10

29)
Try the bracket method to plot the solution to a problem you just did: #10 on
page 314

30) page 324
#2

_{ }