Welcome to Math 244, Sections 01--03. I hope that you will enjoy the course and do well.
The name of your recitation instructor is Charles Wolf, and his email
address is ciw13@math.rutgers.edu.
Grading for the course will be 100 points for each of Exams 1 and 2,
200 points for the Final Exam, 50 points recitation quizzes, 25 points
for Maple, and 25 points for homework.
There will be a separate grader for Maple Labs. Maple Lab 0 will not
be graded but must be handed-in.
You can find the syllabus and information about the textbook on the
course webpage. This includes a rough idea of when Exam 1 and Exam 2
will be given.
In general, your homework assignments will be what is listed on the
course webpage under "Suggested Homework Assignments Math 244", but
you should always check below, on this webpage not on the course
webpage, for the exact assignment, since there may be some
differences.
Homework due Thursday, Jan. 31
1.1: 1,2,5
1.2: 1b,2a,7,9,12
1.3: 1,3,4,6,8,13,14
2.1: 2,6,8,13
You should expect a quiz on this material in recitation on Thursday,
Jan. 31.
Homework due Thursday, Feb. 7:
2.2: 3,8,9,17
2.3: 7a,8,13,16,20,23abcd,30abd
2.4: 1, 20
MAPLE 0 is due on Feb.7 at recitation.
EXAM 1 IS SCHEDULED for Monday, March 4 at lecture. The material will
be everything up to and including section 3.2.
Homework due Thursday, Feb. 14:
MAPLE LAB 1 is now assigned and due at recitation on Thursday, Feb. 21
2.5: 2,9,10,22
2.6: 2,3,5,8,9,14,19,26
2.7: 2
Homework due Thursday, Feb 21:
8.1: 2ac,16,19
8.2: 7a only for t=0.5 and t=1.0
8.3: 7a only for t=0.5
PLUS solve the second order initial value problem 2ty''-y'+ (1/y') =0
with y(1) =3 and y'(1) =2.
Homework due Thursday, Feb. 28:
3.1: 3,6,10,16,18,21
3.2: 2,6,7,11,13,24,29,33
Suggested Review Problems: Here is a list of some typical problems to help
you study for Exam 1. The list may NOT cover every type of problem that
will be on the exam, and not every problem on the list will correspond to
an exam problem. There are 11 problems on the list, 9 from the book and
2 others that are written out below.
From the book: 1.2:14; 2.1:11(c); 2.2:6,26; 2.5:3; 2.6:30; 3.2:28(a);
3.3:34,38.
Two more:
(1) Carefully sketch the direction field for the equation y' = t+y^2-1.
Identify the types of curves which are level curves, e.g., are they conic
sections? Sketch and label a few specific level curves. Carefully sketch
the solution curve y(t) with y(0)=1, but don't try to solve the
corresponding initial value problem. Does there exist a unique solution
curve with y(0) =3? Explain your answer!
(2) Consider the initial value problem y' = 10 sin 5y, y(pi/10) =1. Don't
try to solve the problem. Use Euler's method with step size pi/10 to
estimate y(pi/5) by hand. What step size guarantees that the local error
is at most 10^{-4} on the interval pi/10 < t < pi/2? For this step size,
what is the order of magnitude of the local error for the Runga-Kutta
method? (Answer to last part: (500)^{-5}. )
SPECIAL REVIEW SESSION: Friday, March 1, ARC 204, beginning at 12:30.
MAPLE LAB 2 IS NOW ASSIGNED AND DUE AT RECITATION ON THURSDAY, MARCH 14.
Homework due on Thursday, March 7:
3.3: 1,4,7,16,17,22,24
Homework due Thursday, March 14:
5.4: 2,4,8,9,15,35
3.4: 2,11,14,19
Homework due Thursday, March 28:
3.5: 4,10,17,20a,25a
Note that 20a and 25a ask you only to find the FORM of a prospective
particular solution. You don't have to find the specific numerical values of
the various constants.
Mr. Wolf will not be holding office hours on either Wednesday, March
27 or Monday, April 1.
Homework due Thursday, April 4:
3.6: 1,7,15
3.7: 1,4,5,9 PLUS
Find A, B if y(t) = e^{-2t}(A cos 3t + B sin 3t) and y(0) = -4, y'(0)
= -1. Write y(t) in the form R e^{-2t} cos(3t - delta). Carefully
sketch y(t) for all t >= 0. Find the first positive t at which y(t) is
0. Find the quasiperiod of y(t). (Note that y(t) is NOT periodic
because the exponential e^{-2t} is present. The quasiperiod is defined
to be the period of the cosine factor, cos(3t - delta). The quasiperiod
does lead to a fixed distance between the roots of y(t) even though
y(t) is not periodic.)
EXAM 2 IS SCHEDULED FOR MONDAY, APRIL 22.
MAPLE 3 IS DUE AT LECTURE ON MONDAY, APRIL 15.
Homework due Thursday, April 11:
3.8: 1 including a careful graph, 2,5,7a,11a PLUS
Consider the spring-mass system 4y''+4y'+y =0 with y(0)=1 and
y'(0)=2. Find the solution y(t) and carefully sketch its graph for t
>= 0. Next, do the same if the initial conditions are changed to
y(0)=1 and y'(0)=-2. For these problems, does the spring cross its
equilibrium position y=0? If so, how many times does it cross, and
when does it cross? Be sure that your graphs indicate this.
4.2: 11,19,23
Here is a list of some review problems for Exam 2. Exam 2 will cover
all the sections listed below.
3.3: 18
5.4: 3, 14
3.4: 15ab, 28
3.5: 6, 19a
3.6: 5
3.7: 2 including a careful graph, 6
3.8: 3 including a careful graph
4.2: 14
5.1: 7, 28
5.2: 7
Homework due Thursday, April 18:
5.1: 1,6,8,9,14,15,18
5.2: 2,3,5,9,15a
Exam 2 will not include section 5.3.
REVIEW SESSION FOR EXAM 2: FRIDAY, APRIL 19, 1:30--3:30, SEC 203
OUR FINAL EXAM: Wednesday, May 15, 12:00--3:00 PM in our usual room.
Some Suggested Problems for Recitation on Thursday, April 25 (Not to
be handed-in, but typical problems that you should study):
7.2: 22
7.5:15,16 with these extra questions:
If we denote the first and second components of the solution vector
x(t) by x_1(t) and x_2(t) respectively, find the limits lim_{t \to +
infty} x_1(t), lim_{t \to + \infty} x_2(t), lim_{t \to - \infty}
x_1(t), lim_{t \to - \infty} x_2(t), lim_{t \to + \infty}
[x_2(t)/x_1(t)] and lim_{t \to - \infty} [x_2(t)/x_1(t)], if these
limits exist.
Homework due Thursday, May 2:
9.1: 1(a), 2(a) plus for each of these, find each of the 6 limits listed
above in the extra questions about 7.5
7.6: 3(a), 9 plus this: for number 9, which (if any) of the 6 limits
above exist, and if they exist, what are their values?
SOME REVIEW PROBLEMS FOR THE FINAL EXAM.
The problems below do NOT cover all types of problems that will be
tested on the final exam. Most of them are about material we covered
after Exam 2. You'll need to review the entire course. It might be a
good study idea to retake both Exam 1 and Exam 2, and also to rework
the review problems for both exams.
(1) One solution of t^2 y''-t(t+2)y'+(t+2)y = 0 for t>0 is y(t)
=t. Use the method of reduction of order to find a second solution
y_2(t) so that the pair y_1(t), y_2(t) form a fundamental set of
solutions. [Answer: t e^{t}]
(2) Do 5.2: 6abc.
(3) Find the general solution of the 5th order nonhomogeneous
o.d.e. y^(5) -81 y' = 9. Here y^{5} stands for the 5th derivative of
y. Start by finding the general solution of the corresponding
homogeneous equation y^(5) - 81 y' =0. In order to find a particular
solution of the nonhomogeneous equation, you can use the usual sort of
ideas related to the method of undetermined coefficients. Having found
this, what is the general solution of the nonhomogeneous equation?
(4) Find the eigenvalues and corresponding eigenvectors for the 2x2
matrix whose top row is (1 2) and whose bottom row is (2 1).
(5) Consider the first order linear system x_1' = x_1 + 2x_2, x_2' =
2x_1 + x_2 for the functions x_1(t) and x_2(t). Note that the
coefficient matrix for this system is the matrix in the preceding
problem! Find the solutions x_1, x_2 which satisfy the initial
conditions x_1(0) = 2, x_2(0) = -1. Find lim_{t \to - \infty}
x_2(t)/x_1(t), lim_{t \to + \infty} x_2(t), and lim_{t \to - \infty}
x_1(t).
(6) Find both the complex form and the real form of the general
solution of the first order linear 2x2 system whose coefficient matrix
has top row (1 4) and bottom row (-2 -3). Express your answers in
terms of column vectors.
REVIEW SESSION FOR OUR FINAL EXAM: The review session will be held on
Monday, May 13, 2:00--5:00 PM, in SEC 202.
Please note the corrected answer for review problem 1, and the change
from +81 to -81 in review problem 3.