2) page 18 # 24, 26 and 36 plus page 43 # 1-4 Some remarks: On problem 26, tMAX
is approximately20.4 seconds.
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,
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
pg 73 # 33, 35, 37, 39 Do
these by the systematic way introduced in class, please.
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.
pg 111 # 1, 2, 5, 6
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
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
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.
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
page 147 # 18 and 20
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.
October 13 th
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
#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).
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
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….
page 206 # 2, 3, 4 plus complete the solution of the equation: y” + 9y =0
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.
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.
With a minimum of work, find the formula for the Laplace transform of the
second derivative of f(t), as discussed in class.
page 287 # 2, 3, 4,5
page 314 # 1, 2, 3, 5, 6
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)
page 324 #2 plus: solve x” + 9x = 4 times (delta ^ 6 Pi) with
x(0) =1, x’(0) = 0
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.
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?
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 )
3$) No assignment