Math
3230.A /
Fall
2024
Homework
Ais stated in the Syllabus:
HW
for a section is assigned on the day when we have finished covering
that section in class.
HW # 1
Sec. 1.2: ## 5,
7, 8, 9, 13, 14, 21, 23, 3, 10.
Sec. 1.3: ## 5, 1, 2, 4, 10.
Additionally, for ## 1, 4, sketch the solution
of the DE with initial condition y(0) = 1/3.
See the Note below and do what it says. Make sure that the solution obtained with DSolve,
as per the Note, agrees with your
direction field, as illustrated in Ex. 4 of Lecture 1.
Here is a scan of the pages containing the homework problems: sec1.2, sec1.3
Notes for # 1.2.23: 1) Start with DE y'' = -g. What will be the initial condition for y?
What will be the initial condition for y' (i.e., for the velocity)?
Solve the initial value problem extending the steps of Ex. 1 in the
posted Notes. How many arbitrary constants does the general solution
of the DE have (i.e., before you have used the initial conditions)?
2) To answer the question about the impact time:
What is the condition on y(t) that defines the impact?
Hint for # 1.2.10: Proceed similarly to # 23.
Hint for # 1.3.4: What are the
equilibrium solutions? Now follow an example in
the posted Notes.
Note for the Additional
Assignment for ##
1.3.{1,4}:
We have not yet learned any techniques of solving DEs and IVPs.
Therefore, you need to use command DSolve in the computer program
Mathematica to find this solution for you. The
syntax of this command
is found when you enter its name into Mathematica's Help.
Answer for #1.2.8: any k
Answer for #1.2.14: y0=-1, g(t) = sin(t)+cos(t)
Answer for #1.2.10: y = (t^3/3) + c1*t^2 + c2*t + c3
Answer for #1.3.10: e.g., y' = 2y-y^2
HW # 2
Sec. 2.1: ## 1,
3, 5, 6, 9.
Sec. 2.2: ## 1, 3, 4, 7; 11, 14, 20,
21; 25; 27; 28; 36, 37,
39; 29(c); 41.
Sec. 2.3: ## 19, 21.
Sec. 2.9: ## 18(a,b).
Scanned pages from the textbook:
Problems for secs. 2.1--2.4,
Answers for secs.
2.1--2.4; Pages 25,26; Pages for sec. 2.9.
Note for all
problems requiring the solution of a DE or IVP:
You must obtain
your answer using one of the techniques illustrated in class.
You may check
your answer using command DSolve
in Mathematica.
The syntax of this command is found when you enter its name into
Mathematica's Help.
To encourage you to use this tool, I will not be listing
answers to even-numbered problems
that require solution of DEs or IVPs.
Note about solving homogeneous
DEs in this HW (and in future ones):
Instead of using the general formula for the solution right away,
begin by examining if this DE has already been solved in topic 4 of
Lec. 2.
If it has been, then you can use the solution provided there.
Hint for # 2.2.28: Examine the sign of y' in
each DE.
What does this sign tell you about y(t) increasing or
decreasing?
(Review related comments on p. 2-10 of the posted notes for Lec.
2.)
Hint for #
2.2.36: Of course, you need
to first solve this non-homogeneous IVP.
What will allow you to find the integral there is a special relation
between g(t)
and p(t) (and
hence P(t) ). To
be able to use
that special relation, you will also need to review the Chain Rule and
the Fundamental Theorem of Calculus from Calculus I.
Hint for #
2.2.37: Again, solve
the IVP first. To answer the question of the problem,
review the behavior of the exponential function as t->
Infinity.
Hint for #
2.2.39: Find the place
in Lec. 2 where this DE was solved.
Note for # 2.2.29(c): We basically did #29(a,b) in
topic 5 of Lecture 2.
Use the formula for the
solution derived there.
Note for # 2.2.41: This is a
problem about the material of pp. 25-26,
which you were asked to read on your own.
Note for all problems in Secs. 2.3 and 2.9:
Use the general solution
derived in topic 5.
Notes for all problems in Sec. 2.3:
1) Newton's Cooling Law
was stated in sub-Ex. 2(b) in Lecture 1.
It is also stated in Eq. (4) on
p. 36 of the textbook; note that in
all problems of interest to us, S(t) = S0 (i.e., a constant).
2) While the formula for the solution of the Cooling Law with S(t) = S0
can be found on p. 36, here I expect you to use not it but the solution
derived in topic 5 of Lec. 2.
Note for Sec. 2.9: 1) Read pp. 78, 79 (Drag force, Case
1).
2) The mass of the skydiver will only enter in a combination (k/m), where k
is the air resistance coefficient.
Answer for # 2.1.6: linear
Answers for # 2.2.4,14,20,28,36: See the Note above.
Extra
credit # 1
Assigned: 09/11
Due:
09/18
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Word Problem 1:
(worth
0.3% added to your final
grade)
You have a hot cup of
tea,
initially at temperature T_i,
in a room
of temperatire T_r.
You want to know which of the two methods will
cool the tea to a lower temperature in a given time t.
Method 1:
Right at t=0, you add cold water (with temperatire T_w) and then wait
for the mixture to cool down for t time units.
Method 2:
You wait for t
time units first and then add the same amount of cold
water, with the same temperature, as in Method 1.
Assumption for both methods:
When you add A
parts of water to (1-A)
parts of tea with temperature T,
the temperatire of the mixture becomes A*T_w + (1-A)*T.
- Does your answer depend on a relation between T_r and T_w?
- Does your answer depend on A
(i.e. on the amount of cold water added)?
Clarification:
You may work out an answer for specific values of parameters of this
problem.
However, full credit will be given only if present your work for
general values.
Word Problem 2:
(worth
0.2% added to your final
grade)
Let P(t) be the
performance level
of someone learning a skill as a function
of a training time t. The graph of P(t)
is called a learning curve.
The
following IVP is a reasonable model for learning:
dP/dt = k*(M -
P(t)), P(0) = 0,
where k, M
are some positive constants.
Two new workers were hired for an assembly line. The above learning
model
is assumed to apply to both of them. Here P(t) denotes the number of units
that either of them processes at time t. For the following you should assume
that t is measured only in whole hours.
Jim processed 25 units of product
during the first hour and 45 units during the
second hour. (Thus, e.g., P_{Jim}(2) = 45.) Mark processed 35
units
during the first hour and 50 units during the second hour. Which worker
is capable of producing more product units when they both become
professional in their job?
HW # 3
Sec. 2.1: ## 11(a),
13(b), 16.
Sec. 2.2: ## 33, 31.
Word Problem 1:
Prove part (b) of the Superposition
Principle.
Word Problem 2:
(a) Prove the statement of Note 1 on p. 3-6 of
posted Lecture 3:
If y_1 and y_2 are two solutions of the same non-homogeneous DE,
then (y_1 + y_2)
is not
a solution of that DE.
(b) Let y_p be a particular solution of a
non-homogeneous DE.
Use the method of part (a) (which is the same method as that used
to prove part (a) of the Superposition Principle) to determine
for what value(s) of constant C,
C*y_p is also a solution
of that DE.
(c) Let y_p be a particular solution
of a non-homogeneous DE
and let y_h be a solution of the coresponding homogeneous DE.
Use parts (a,b) above and the result of Word Problem 1 to determine
for which values of constants C_h
and C_p the
function
y = C_h * y_h + C_p * y_p
is also a solution of the same non-homogeneous DE:
(1) C_h=0, C_p=0;
(2) C_h=1,
C_p=1;
(3) C_h=0, C_p=1;
(4) C_h=1, C_p=0;
(5) C_h=0, C_p= -1; (6) C_h=
-1, C_p=0;
(7) C_h=1,
C_p=1;
(8) C_h= -1, C_p= -1;
(9) C_h=0, C_p=2;
(10) C_h= 2, C_p= 0.
Word
Problem 3:
1) Guess a particular solution of the DE in
# 36 of Sec. 2.2; [it's simple!]
2) Find the general solution;
3) Use the result of 2) to solve the IVP in # 36.
Note for #
2.1.13(b): Recall that Thm.
2.1 imposes conditions not only on p and g.
but also on t_0.
Note for #
2.1.16(a): Find the solution
of the DE in one of the examples in Lec. 2.
Note for # 2.1.16(b,c): Something similar was
done in class in Lec. 3.
Hint for ## 2.2.{33,31}: Use part (b) of the Superposition
Principle as follows.
First, deduce yh(t) and yp(t) from the solution given in
the problem.
Second, find p(t)
by comparison of yh(t)
with the solution of the
homogeneous DE as found in topic 2 of Lecture 2.
Finally, find g(t)
from the particular solution yp(t)
by substituting it
into the DE, where now
you know both p(t) and yp(t).
Answer for #2.1.16: (a) C=1, r = 3; (b) (-infinity,
0); (c) all t
Answer for WP #
2(b): 1
Answer for WP # 2(c): 2, 3, 7.
HW #
4
Sec. 2.4: ## 3;
13(a,c), 15. (Pages from the textbook that you will need
are posted in HW # 2.)
Sec. 2.3: ## 19, 21, 16, 17 and Sec. 2.4.14;
Sec. 2.3: ## 2, 3, 1;
7(a,c), 6.
Sec. 2.9: ## 4, 1.
General Note: One can apply the solution found in
topic 5 of Lec. 2 (p. 2-11) to some
(but not all)
of the problems in this assignment.
Note for #
2.4.3: Even though this problem is about a
bacterial population, its solution
should mimic that of Ex. 1 in the Notes for radioactive decay.
Notes for ## 2.3.{19, 21, 16,17} and 2.4.14:
1) The first four of these problems are on Newton's Cooling law (see
below), while
the third one is on radioactive decay. They are assigned together
because they
lead to an equation of the same form. (You will need to realize that
when working
on # 2.4.14.) Therefore, I expect you to solve all of them using the
same form of
the solution, given by the last equation on p. 36. This solution can be obtained
by the method of topic 5 of Lec. 2 and, in fact, was given there
explicitly.
2) In some of these problems, you'll need to find the constant k.
See the Note for # 2.4.3 above.
3) You had to do ## 2.3.{19,21} in HW 2 using the method of topic 5 of Lec. 2.
Here I expect you to use the aforementioned formula from p. 36.
Note for ## 2.3.{19, 21, 16,17}: Newton's
Cooling Law was stated in sub-Ex. 2(b) in Lecture 1.
It is also cosidered on
pp. 35--37 of the textbook.
Clarification for #
2.3.17: It is meant that the oven and the pie, referred to
in the second sentence
of the problem, remain identical to those in the first sentence.
This implies that the constant k
in Newton's Cooling Law is
the same
in both sentences.
Hint for #
2.4.14: Write down the equation for dQ/dt first. Note that the phrase "added at a
constant rate" translates into a term "+R" added to the r.h.s. of the
"standard" radioactive decay equation, found at the beginning of topic 2
of Lecture 4.
(I don't just write this
equation here because I want you to review, based
on that Lecture, how you set up rate equations:
Rate of change of X = (Amount of X added per unit time) -
(Amount of X subtracted per unit time).
If you are still unclear, look at Example 2 on p. 43 or at the formula
in # 10 on p. 46 of the textbook.)
Clarification for # 2.3.3: The last sentence simply
asks
you to convert the answer (m^3/min)
into a dimensionless one by dividing the former by the volume of the
room.
Notes for Sec.
2.9: 1) Read pp. 78, 79 (Drag force, Case 1).
2) You will need to solve an equation of the same form as in ## 2.3.{16,17} and 2.4.14;
therefore, you should try to apply the same method here as you have
practiced there.
Hint for # 2.9.4:
You will need to
first solve for v(t) given the information supplied by the problem.
Now, at the maximum height, what is the velocity of the projectile?
Answer for #2.4.14: 50*(ln2/3)*(2 -
exp(-A))/(1-exp(-A)), where A = 2*ln2/3
Answer for #2.3.16:
10*ln(139/24)/ln(139/109)
Answer for
#2.3.2:
50*ln10
Answer for
#2.3.6: (a)
400 min, (b) 7.5 lb, (c) 10 lb at 200 min
Answer for
#2.9.4: t =
ln( (v0+mg/k) / (mg/k) ) *(m/k)
Answer for
#2.9.1:
Using the solution at the top of p. 79, we obtain: T = (m/k)*ln(2).
Extra
credit # 2 (each
Word Problem is worth
0.25% added to your final
grade)
Assigned: 09/16
Due:
09/23
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Word Problem 1:
Consider the following
variation of # 2.4.13(b).
(a) Let the ratio of C-14 to C-12 be known with a small error of p%.
I.e., it is 30%*(1 + 0.01*p). Assume that the half-life T is still known
exactly, i.e., T = 5730 years.The uncertainty of the ratio of
C-14 to C-12
creates an uncertainty
in the estimated age, A.
What is the percentage error of this uncertainty? Does it depend on A?
Hint:
At some step of your solution, you will need the first term in the Maclaurin
expansion of ln(1+x) when x is very small. (You
learned this in Calculus II.)
(b) Similarly, if the half-life T
of C-14 is known with a small error of p%,
what uncertainty, in percent, does it create in A? Does it depend on A itself?
Hint:
Use the technique of linearization, which was
studied in Calculus I and III for
functions of one and two variables, respectively. (Here you only need
one variable.)
That is, for a function f(x), relate the
change/error/uncertainty in the function,
(delta f), to (delta x) when the latter is small. Then the
percent uncertainty
in f is (delta f)/f.
Word Problem 2:
Read pp. 78, 79, 82 and
Examples 2 and 3 of Sec. 2.9. Apply the technique of p. 82
and Examples 2 and 3 to # 2.9.16.
Hint: Define
a new variable z=v^2
and solve the equation for z
using a
technique from an
earlier Lecture.
HW # 5
Sec. 2.6: ## 1, 3, 4, 7, 11, 23(b), 9;
27; 24; 30.
Sec. 2.5: ## 11, 12.
Word problem:
(a) Solve the IVP y' =
-(1+y^2), y(pi/4)=-1.
(b) Following the General
note 1 in Lecture 5, explain why this
solution
is different from that of #2.6.7.
A friendly reminder: Please read and follow the
"Note
for all
problems requiring the solution of a DE or IVP" in HW 2.
Hint for #
2.6.3:
You will need to solve a quadratic equation.
Note for #
2.6.7:
You may need look up the integral over y in a table of integrals,
in a Calculus II textbook, or in Mathematica.
This integral will come up several times in this assignment
and in the future.
Note for ## 2.6.{23(b),9}: These problems (with
different coefficients) will occur
in several applications considered in this course;
therefore it is important that you remember how to
solve these problems.
Hints for ## 2.6.{23(b),9}: 1. Use partial fraction
expansion for the y-integration.
If you do not
remember steps of this technique from Calculus II,
you MUST review it.
We will use it multiple times in this course.
2. After the integration,
you will end up with an equation of the form
(y +
a)/(y+b) = f(t),
where a
and b are some
constants. To solve
that equation,
first multiply both sides by the denominator. Then collect terms
that contain y on
one side and, finally,
solve for y.
Hint for #
2.6.27:
1. First, answer their first question ("How long..."), and only after
that
proceed to the second half of this problem ("Recall that...").
2. After you solve for Q as a
function of t and
the constant C
(which you'll have found from the initial condition), you will likely
end up with an
expression of the form
similar to Q = 1/(a + b/c)
for some a,b,c. So,
you'll have a fraction in the demoninator.
It is
conventional to avoid fractions
in denominators (in most situations),
and so you'll need
to simplify that
expression to make is look like the
usual fraction,
i.e. A/B for some
A,B, which are related to a,b,c.
This will make your solution for Q(t) look much neater.
Hint for
#
2.6.24:
Complete the square and then reduce
the problem to something
very similar to #2.9.7 following the trick of topic 5 of Lecture 2.
(If you do
not remember how to complete the square,
you must find it in a textbook or online and review it on your own.)
Hints for
#
2.6.30:
1.
We solved the first equation in
class;
You solved an equation very similar to the second equation
in one of the earlier problems in this HW (use the General Note 1
from Lecture 5 to account for the difference between the solution
of this equation and the one you have solved earlier);
You solved a very similar equation to the third equation in yet another
earlier problem in this HW. (The aforementioned General Note 1 can be
useful here also to deduce the qualitative behavior of the solution.)
2. To verify your
answers, follow the "friendly reminder" above in this HW.
Hints for #
2.5.11:
1. The end of the second sentence in this problem could be made
clearer
if one uses y-bar instead of y:
"... the initial value problem (y-bar)' = f(y-bar), y-bar(1) = 2."
2. To clearly see the connection between this problem and one done
in class, I recommend that you sketch y(t) and y-bar(t) in the same
graph.
Answer for # 2.6.4:
-1/(1+t^2)
Answer for # 2.6.24:
tan(pi/4 + t) - 1
Answer for # 2.5.12:
(a) 2; (b) 0.
Special Extra
credit:
Project 1 (worth 1% added to your final
grade)
Assigned: 9/16
Due: 09/30
Path of a pursuer chasing
a prey (Curve of pursuit)
(source: "Fundamental
of Differential Equations" by R.K. Nagle, E.B. Saff, A.D. Snider; Chap.
3)
Do Parts (a) -- (f) of
Project C in the above file. Additional 0.2% will be added to your
grade if:
- You get credit for the Project, and
- Solve its Part (g). (Hint: Use L'Hospitals
Rule on the 2nd and 3rd terms, taken together, in Part (f).)
Read and follow instructions
for submission of Extra Credit Projects
(which are different
from the instructions for non-Project Extra Credit assignments).
HW # 6
Sec. 2.5: ## 5, 9.
HW # 7
Sec. 2.5: ## 13(a), 14(a), 17(a), 19(a).
Sec. 2.6: ## 35, 37, 36; 38, 33.
Sec. 2.7: ## 13, 1, 5, 9, 23.
Word Problem: Find the general solution of the
given DE:
y' = (t^2 -
y^2)/(3ty)
A friendly reminder again: Please read and follow
the
"Note for all
problems requiring the solution of a DE or IVP" in HW 2.
Note for ## 2.5.13(a): Even though this is a separable
equation, here you need to solve it
as a Bernoulli equation to practice this new concept.
After you do so, compare your solution (both the overall process
and the answer) with that of ## 2.6.23(a) in HW 5.
Note for ## 2.5.14(a): This is also both a
separable and a Bernoulli equation.
You may do this problem by either method. If you choose to use
the separable equation method, follow the steps of your solution of
## 2.6.{23(a),9} in HW 5.
Hint for # 2.5.19(a): You will need to
integrate by parts at some point.
Hint for # 2.6.37: After
you transform the DE to a separable one,
use the partial fraction expansion.
Hint for # 2.6.36:
To do the integral over the
dependent variable, first get rid
of the improper fraction. If you do not remember
what this means
and how to do it, review the ground work that needs to be done
before one does partial fraction expansion. (See the related Note
for HW 5.)
Hint for # 2.6.38:
After you make a transformation, compare the resulting equation
with that in # 2.5.13(a).
Hint for # 2.6.33:
Divide both the numerator and denominator of the r.h.s. by t.
The equation will then take the
form of one of the "special cases"
considered in class.
When you integrate
over the dependent variable,
you need to break the integral into two terms, where each term is to
be done by its own
integration method (both being very simple).
Note for ALL problems in Sec. 2.7:
1) Study pp. 7-4 through 7-6 of the posted Notes first.
2) Recall that to (re)learn the method of finding H(t,y),
you need
to read the non-numbered Example on p. 66 and then the more
general Example 2 on p. 67.
Keep in mind that the book's notations M(t,y) and N(t,y)
stand for the quantities denoted m(t,y) and n(t,y) in the posted Notes.
Note for ## 2.7.1, 5, 9, 23:
Recall that while you will be finding a function H(t,y)
which is given
by some expression of t
and y, the solution has
the form:
"that expression of t and y " = C,
and not
"that expression of t and y " = H(t,y)
(the latter is just a tautology, not an implicit solution).
Hints for the Word problem: 1) Divide both the
numerator
and denominator of the r.h.s. by
t^n,
where n is such
as to make the DE appear in the form of one
of the special cases considered in Lecture 7.
Alternatively, you may just
divide each term in the numerator by
the denominator.
2) The integration over the
dependent variable can be
done with a
simple technique, covered in Calculus I.
Extra
credit # 3 (each
Word Problem is worth
0.25% added to your final
grade)
Assigned: 09/27
Due:
10/04
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Word Problem 1:
This extends the idea of topic 3 of
Lecture 7.
Find the general solution of the DE y' = (y - 3t + 6)/(y + t + 2).
Hint: If you attempt to follow the steps that you have
followed for one of the
problems in the regular assignment, you'll soon find that terms 6/t and 2/t
get in your way. Therefore, the constants 6 and 2 need to be eliminated
from the original equation before you can apply one of the earlier
techniques.
To eliminate those terms, you need to make a transformation of
both t and y:
t = u + a, y = v + b,
where u and
v are the new variables
and constants a
and b are chosen
so that
the numerator and denominator appear as v + c*u and v+d*u, respectively,
for some constants c
and d.
Word Problem 2:
This shows that while the
Riccati equation cannot be solved in general,
it can still be solved when an extra piece of information is available.
Consider a Riccati DE y'
= x^3(y - x)^2 + (y/x).
(i) Verify that u(x) = x
is its solution.
(ii) Substitute y = u(x) + v into the given
Riccati DE and obtain a DE for v.
(iii) Solve the latter DE by a
technique from Lecture 7.
This method of "solving" the Riccati equation can be stated in the
general form:
Suppose
u(x) is a known solution of a Riccati equation. Let v = y -
u(x). Then the DE satisfied by v
is solvable. Hence,
knowing one solution of the Riccati equation, one can find its general
solution.
Note: Strictly speaking, it may be unclear why
we claim to be able to find the general solution of the
Riccati equation as
opposed to just another
particular solution of it. The fact that we are indeed
able to find the
general solution follows from an observation that an equation satisfied
by some
variable related to v
is linear, and for linear equations we can always find the general
solution.
The remaining Word Problems are based on topic 2 of Lec. 7, which (the topic) we skipped in class.
Read the notes for that topic and apply its technique to find the general solution of each of the following DEs:
Word Problem 3:
y' = 1/(t + e^y)
Word Problem 4:
y' = y/(y^2 - t)
Hints for Word problem 4: Divide
both the
numerator
and denominator of the r.h.s. by y,
so
as to make the DE appear in the form of that considered in topic 2.
HW # 8
Sec. 2.8: ## 1, 2, 3; 4, 5, 6, 7;
18, 19.
Hint for # 2.8.18: See Eq. (7) in Sec.
2.8 and use the exact solution of Eq. (1) there.
(This exact solution is given by Eq. (5) in the book and was also
quoted in Lecture 8.)
Answer for # 2.8.2: 10*ln(4.4)
Answer for # 2.8.4: 1/4, 3/4; 3/4 is the
stable one
Answer for #
2.8.6: 1/2; P (t) ->
-Infinity as t increases
HW # 9
Assigned: 09/28
Sec. 2.10: ## 3, 5 with the following additional
instructions:
1) Do not
compute y_3
and e_3.
2) In addition, repeat the assignments for h = 0.05 and compute y_1, y_2, y_3, y_4
for this new h,
as well as e_1, e_2, e_3, e_4.
Verify that e_4
for h = 0.05
approximately equals half of e_2 for h = 0.1
(note that both errors are computed at the same value of t = 0.2).
3)
Sketch together the exact solution and the two approximate solutions
(i.e., those
for h = 0.1
and h = 0.05).
Answer these questions:
a) Which approximate solution is closer to the exact one?
b) Are both approximate solutions on one side or on
opposite sides of the exact solution?
c) Does your answer to question b) agree with the construction of
the approximate
solution using the concept of slope (see p. 9-4 of posted Lecture 9 and
p. 91 of the textbook)?
d) Does it agree with the magnitude and sign of the errors that
you have obtained in item 2)?
A Matlab code
by which you can check your work above.
It is written for #2.10.3, but can be easily adopted for # 2.10.5.
Also, if you don't know Matlab but know Python or Julia, you should be
able
to understand the code and, if needed, make your own copy in another
language.
HW # 10
Sec. 3.1: ## 11; 9, 10, 13, 14; 3, 7.
Word Problem 1:
(a) Verify that x = sin (w*t)
is a solution
of the linear oscillator model (p. 10-4 of posted notes).
(b) Same question for x = C1*cos(w*t) + C2*sin(w*t),
where C1, C2 are any
constants.
Word Problem 2:
Consider a mass suspended on a vertical
spring. Similarly to the horizontal spring
considered in class, one can show that the equation of motion in this
case is
y'' = -w^2*y - g,
where g is the
specific gravity and w
is the same as for the horizontal spring.
Use the trick of topic 5 of Lecture 2 and the material of Lecture 10 to
obtain the
general solution y(t)
to the above equation.
Hint
for # 3.1.11: In solving these problems, you
need to solve the DE for y''
(i.e., rewrite the DE as y'' = ... )
at that value of t where you know all other terms in the DE.
I'm being vague here because I want you to
look at the given IVP
and realize what value of t that is.
Note for # 3.1.13:
Refer to the equation in # 3.1.12 or to Eq. (3) in Sec. 3.1.
Notes for # 3.1.14: 1.
Ignore the dimensions of the cylinder. They are irrelevant to the
questions being asked.
2. This problem refers to the picture on page 115. (This is quite
unclear from the problem's statement.)
3. For part (b), you will need to think about the amplitude of the
sinusoidal function. For example, what is the amplitude of 2 sin x?
Answer for # 3.1.10: C1 = 1, C2 = -1.
Answer for # 3.1.14: (a) y0 = 0, T =
2; (b) omega = Pi, y0' = 6*Pi.
Note for WP
2:
The
answer is so simply related to the topics indicated that I don't want
to list it.
Extra
credit # 4 (worth
0.3% added to your final
grade)
Assigned: 10/07
Due: 10/14
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 3.1: # 12. This problem is based on the description found on pp. 108, 109 in the book.
Recall that to receive credit, you
must explain every step your solution.
HW # 11
Sec. 3.2: ## 1, 3, 6, 7, 9, 10, 12, 13; 21,
20; 17, 16.
Word Problem:
In
Example 3 of Sec. 3.2 (see p. 119), W(t) crosses zero (at t=0).
Why does this not contradict the first Note on p. 11-6 of Lecture 11?
Hint for # 3.2.21: Use
Eqs. (7) in the posted Notes.
Hint for # 3.2.16(b): See Example 1 of Sec.
3.2.
If you do not
remember the trig formula they used, review it!
Formulas for sin(x +/- y)
and cos(x +/- y)
will be used often.
Hint for #
3.2.20: At the end, you
will need to use one of the formulas mentioned
in the previous hint.
Answer for # 3.2.6: (b)
0.5*exp(t/2), (c) Use DSolve in Mathematica.
Answer for # 3.2.10: (b) - ln3 /
t, (c) Use DSolve in Mathematica.
Answer for # 3.2.16: (b) c1 =
1/(2*sqrt(2)), c2 = 1/sqrt(2).
Answer for # 3.2.20:
alpha is not Pi*n/2, where n is an integer.
HW # 12
Sec. 3.3: ## 1, 3, 5, 7, 9, 11, 13;
17; 18; 19, 20.
Additional Assignment
for ## 1, 3, 5, 7, 9, 11, 13, 20:
Sketch the solution on the whole line.
(For the sketch in # 20,
let m = k = 1, x0 =
0, v0 = 2.)
Make
sure that you can explain
whether your graph crosses the t-axis
and if so,
then at what t=tc
(positive or negative) this occurs.
That is, you must compute tc (see the Hints below) for each of the problems
where the graph intercepts the horizontal axis.
In addition, explain why the graph is positive on one particular side
of tc
and negative on
the other (that is, why one exponential term dominates for
t --> +Infinity and why the other term dominates for t --> -Infinity).
Note for solving certain quadratic equations:
If you are solving a "full" quadratic equation a*L^2 + b*L + c =
0,
you have no choice but factor it by inspection or use the quadratic
formula.
However, if either b = 0 or c = 0, there is a simpler
way. (Let a=1 for simplicity.)
b = 0: L^2 + c = 0 => L^2 = -c => L = +/- Sqrt[-c].
c = 0: L^2 + b*L = 0 => (L + b)*L = 0 => L = -b and L = 0.
Note for the Additional Assignment above:
The emphasis of this assignment is on your being able to
sketch the solution (given by your formula) by hand.
Once you have sketched it (see the Hints below), you
should verify your hand-drawn sketch by plotting it
using some software. (E.g., Mathematica, or its free
web-based version Wolfram Alpha.)
Make sure that the range of your independent variable t
is such that it allows you to see the behaviors of your
graph that are mentioned in the Additional Assignment.
Hints for the Additional Assignment above:
1. To determine the sign of tc (if there is any zero crossing by
the
graph), you need to review for what a one has
ln a > 0 and for
what a it is <
0.
2. To explain why the
graph is positive on one particular side of tc
and negative on
the other, decide which term in the solution
dominates
for t-> +infinity and which, for t-> -infinity.
For
example, if y = 3*e2t - 1, then for t-> +infinity, the term
3*e2t dominates (why?), and the
solution there must be positive.
On the other hand, for t-> -infinity, the term "-1" dominates
(again, why?), and the solution there must be negative.
3. Finally, make sure that both the value and the slope of your graph
at the initial point are those given in the IVP.
Hint for #
3.3.17: (a) Read
the top of p. 124;
(b) If you know x1 and x2, you can find the coefficients of
(x-x1)(x-x2).
Hint for #
3.3.18: To tell
between {A,C} and B, use the method of a very similar problem in Sec.
3.1.
To tell between A and C, you'll need the solution of the IVP.
Note for # 3.3.19: My
two (simple) reasons for assigning this problem are as follows:
1) Sometimes, one can solve a third-order DE by merely reducing it
to a second-order one (i.e., without learning any new techniques).
2) We have learned that general solutions of 1st- and 2nd-order DEs
depend on 1 and 2 arbitrary constants, respectively. Then, on how
many arbitrary constants should the general solution of a 3rd-order
DE depend? Does your work confirm your guess?
Answer for # 3.3.18: (a)=C, (b)=B.
Answer for # 3.3.20: (c) x0 + (m/k)*v0
Extra
credit # 5 (worth
0.15% added to your final
grade)
Assigned: 10/09
Due: 10/16
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 3.3: # 16.
HW # 13
Sec. 3.4: ## 1, 3, 5, 6, 7, 10; 11,
12; 13.
Additional
assignment for ##
3.4.{1,3,5,6,7,10}:
Sketch the
solution on the
whole real line. Make sure
that you can explain whether
your graph
crosses the t-axis
and if so, then at what t
(positive or negative) this occurs.
You must compute this intercept tc wherever the graph crosses the horizontal axis.
In addition,
explain why the graph is positive on one particular side of the zero
crossing
and negative on the other
(i.e., which term in the solution
dominates for t-> +infinity
and which, for t-> -infinity).
See the Note and Hints for the Additional Assignment for HW 12.
In addition, note that term t*e^{at} dominates e^{at} at both
t -> + infinity and -infinity, regardless of the sign of a.
Hint for #
3.4.11:
Essentially, you will need to find two constants, C1 and C2, and alpha
from three conditions:
a) initial value; b) location of the max; c) value at the max.
Note for #
3.4.13: This
problem gives an example of a very simple
Boundary Value problem, or BVP
(as opposed to Initial Value problem),
where the value of the solution is specified at two different points.
More on BVPs will follow in the Extra Credit Project below.
Answer for # 3.4.12: alpha = 0, y0
= 0, y0' = -1/2.
Special Extra
credit:
Project 2 (worth 0.8% added to your final
grade)
Assigned: 10/14
Due: 10/30
1. (worth 0.3%) Do Parts 1 and 2
of Project 3 on pp. 209-210 of the textbook. (Do not do Parts 3 -- 8.)
2. (worth 0.1%) The
main purpose of this Project is to illustrate the main difference
between
linear
Initial and Boundary Value problems with respect to the existence of
solutions. Thus, briefly
describe how conclusions that follow from some
of your results are different from the conclusions
of Theorem 3.1 on p. 111.
3. (worth 0.4%) Even though the
two BVPs considered in item 1 have a very special form, they illustrate
the general
feature regarding how the existence and uniqueness of solution
of the linear BVP
y'' + p(t)
y' + q(t) y = 0
depends on the coefficient q(t).
a) Present your hypothesis about this
dependence on the coefficient q(t).
The general form of your hypothesis should be this:
- If q(t) is so and so, then so and so can happen;
- If, on
the other hand, q(t) is (some other) so and so, and so and so can
happen.
b) Support your hypothesis by presenting
one additional example for each of the cases (so,
one example per case) that you have
identified in your hypothesis. (This statement should
become clear
once you will have stated your hypothesis.)
Note: Your examples, of
course, will have constant coefficients in their DEs,
but they must have a nonzero
p(t).
Read and follow instructions
for submission of Extra Credit Projects
(which are different
from the instructions for non-Project Extra Credit assignments).
HW # 14
Sec. 3.5: ## 1, 2; 3, 7, 9,
11; 13, 15, 19, 21; 23, 24, 25, 26;
27, 28, 29; 31; 33.
For # 3.5.33: In addition, verify that u = Re(y1) and v= Im(y1) are not solutions of the original DE.
Why does this not contradict Theorem 3.3?
Note for #
3.5.1: Here
the book, unfortunately, uses the notations alpha &
beta for
different purposes, thereby creating confusion. Instead of asking
about
"the form alpha + i*beta", it
should have asked about the form "a + i*b".
Note for # 3.5.2:
To check your answers, use command ComplexExpand in Mathematica.
Note for # 3.5.3: In addition
to solving the IVP, sketch your solution via first putting it
in the (R, delta)-form. (Here you will have an answer for
delta in the form
arctan(some # > 0). Keep in mind that 0 < arctan(x) <
Pi/4 for 0 < x < 1
and Pi/4 < arctan(x) < Pi/2 for x > 1.)
You may postpone working on this additional part of the assignment until
you begin working on ## 23--26, which address the same topic. As for
those
problems, here you are also to label the t- and
y-intercepts.
Note for ##
3.5.{7,9,11}:
In addition
to solving the IVP,
make a rough sketch
of your solution.
"Rough" means that you should not use the (R, delta)-form of the solution,
but instead should focus on correctly representing the solution's
envelope
as well as its given initial value and slope.
(Particularly for # 11, I expect you to know what the graph of sin(t) looks like.)
See and follow the Note for Additional Assignment for HW 12.
Notes for ##
3.5.{13,15,19,21}:
1) In
these problems, you are supposed
to do the reverse of
what you did in ## 3.5.{3,7,9,11}. Namely, you are supposed to
recognize the
lambda1 and lambda2 from y and use them
to find a and b: just expand
(lambda - lambda1)*(lambda - lambda2) = 0.
From this characteristic polynomial, you will recover the DE.
2) You should not
solve these problems by substituting y into
the DE and collecting like terms.
3) Also, you do
not need to either
convert the given y into the
(R, delta)-form, nor expand the (R, delta)-form already given in
the problem:
there is simply no need for that in these problems.
4) Finally, the initial conditions are found directly from the
given y(t).
Note for # 3.5.19,21: Expand the cosine first.
Notes for # 3.5.24--26: 1) Remember to do the additional assignment for # 3 above.
2) Before you sketch the full function given in the problem,
sketch its part without the exponential factor(s); e.g.,
for # 24, it will be (cos(t) + sqrt(3)*sin(t) ). In these
sketches, label the
intercepts with the t-axis.
3) Label the intercept with the y-axis.
4) To check your
answers for ## 24,26, use command TrigFactor in Mathematica.
Hint for # 3.5.27--29: Follow these steps:
1) Deduce the period T from
graph.
2) Relate beta to
T using a formula from
Lecture 10.
(There, beta is
denoted by another letter, which, as beta here,
also represents frequency.)
3) Find delta/beta
from graph, as discussed in a class example.
4) Find delta
from 3) and 2).
5) Deduce the amplitude R
from the graph.
6) Finally, use Notes 1) and 2) for ##
3.5.{13,15,19,21} above.
Hint for #
3.5.31: Use
the formula for W
from the top of p. 124.
Answer for # 3.5.28: cos( 3t/2 -
Pi/8)
HW # 15
Sec. 3.6: ## 2, 3, 4, 6(b), 7; 8;
9, 11;
Sec. 3.5: # 32.
Word Problem:
A mass is attached to a spring and to a dashpot with a damping constant
gamma = 60 kg/s. The mass has been set in motion by providing to it some
initial displacement and initial velocity. The resulting displacement as
a function of time is measured to be: y(t) = 0.5e^{-2t} * cos(3t-0.7pi) meters.
Find the spring constant k.
General Note for all problems except # 3.6.8:
Relate your solution to one of the examples worked out in Lecture 15.
Note for ## 3.6.{2,3,9}: In these problems, the book assumes that the downward direction is positive
(which is the opposite of what we assumed in class).
You can follow either convention, but make sure that you consistently
stay
with one and do not switch between the two.
Notes for #
3.6.7: 1) Convert R = 25 cm to meters, since all other units are in meters.
2)The answer in the book seems to be incorrect:
the numbers must have the units [m] and [m/s], not [cm] and [cm/s].
Hint for #
3.6.8: For (c), see the Hint for ## 3.5.{27--29} in HW # 14;
for (d), find a pertinent formula in Lecture 15.
Note for #
3.5.32: As
for any DE or IVP, you can verify your own answer with
Mathematica's or Wolfram Alpha's comman DSolve.
(And, you do not need to know anything
about the derivation of
the
model in question. You just need to
obtain and analyze its solution.)
Hint for the Word Problem:
1. First, by comparing one part of the given solution with the
general
solution for Case 3 in topic 2 of Lec. 15, determine the mass m.
Then, comparing another part of the given solution with the same
general solution, determine k. Here you will need to use a
relation
between the frequencies of the undamped and damped oscillator.
2. Some of the information given in the problem is redundant.
Answer for # 3.6.2:
(a) 3.27*10^3 N/m; (c) 0.07*cos(18.1*t) .
Answer for # 3.6.4:
2000 N/m
Answer for # 3.6.6:
Same.
Discussion: If you
remember/know the concepts of potential, kinetic, and total
energies from Physics, then this answer makes perfect sense. Indeed,
since the initial kinetic and potential enrgies are the same, then the
total energies are also the same. And then, the maximum attainable
potential energies (which occur at the moment when kinetic energy
vanishes) are the same. Since these energies are proportional to the
square of the maximum stretch, as studied in a Physics course,
then those maximum stretches must also be the same.
Answer for # 3.6.8:
(a) 2; (b) f=1/2 Hz, w = pi rad/s; (c) R=3, delta =
pi/4;
(d) y0 = 3/sqrt(2);
y0' = 3*pi/sqrt(2).
Answer for the Word Problem: 195 N/m
Extra
credit # 6
Assigned: 10/23
Due:
10/30
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Motivation: You have studied methods to solve
2nd-order differential equations with constant coefficients:
y'' = f(y, y'),
(1)
where f(y, y') = -cy - by'
and b, c are
constants.
In most cases one cannot solve equations with more complicated forms
of f(y, y').
Here you will see two methods of solving equations of type (1) for some
other
forms of function f(y, y').
Assignment: 1) (worth
0.25% added to your final
grade)
Read the method described on pp.
82--85 of the textbook.
(Note that "our" variable y
is denoted as x
there.)
Apply this method to # 2.9.15.
(Note that if you were to solve this DE relative to t, as we do in class with other equations,
you would have to write its first term as x'', and then the second term would be nonlinear.)
2) (worth
0.4% added to your final
grade)
Read the method described on p. 414 of the textbook (for y'' + f(y) = 0).
The final touch on the method, not mentioned on p. 414, is that you
need to solve
for y'(t), whereby you will end up with a first-order separable DE.
Unfortunately, if one stays with so-called "elementary" functions,
which are those
functions that you have learned so far, one can obtain solutions by
this method only
for a small set of functions f(y).
Apply it for f(y) = y and the initial conditions y(0)=1 and y'(0)=0.
Stay with one sign of the square root.
(This is
a good practice problem since you already know the answer in this case.)
Do all the integrals that you will encounter by hand; in particular, you will need
to use a trigonometric substitution at one of the
steps.
Make sure that the solution you will arrive at is the one you know from
Lecture 10.
3) (worth
0.4% added to your final
grade)
To get an idea how complicated things become when f(y) is not a linear function,
read up to Eq. (8) on p. 415. The pendulum equation was derived on pp.
393-394.
Equation (8) can be solved "in elementary functions" only for E=1.
Obtain such an implicit solution, doing all integrations by hand, as outlined below.
You will need a trigonometric identity for (1+cos(2x)) to begin.
(Note: Here and below, "x" is a placeholder for any variable!)
You will then need to do an integral of sec(x) (for some x) also by hand.
For that, use the identity for cos(x) in terms of tan(x/2);
then use the identity for (1+ tan^2(x)), and then recall the derivative of tan(x).
Finally, use partial fractions for the last integral. You will obtain
an implicit solution
that relates theta (the pendulum's angle) with t.
(worth
0.1% added to your final
grade)
Although you cannot really plot that implicit solution by hand as
theta(t),
you can still plot it as t(theta). Do so and interpret the results.
Make sure to plot only the physically meaningful range of theta-values
(and explain why the range you chose is meaningful.)
(worth
0.1% added to your final
grade)
Explain, providing all relevant details, how the limiting behavior for t -> infinity
follows analytically from your implict solution.
HW # 16
Sec. 3.7: ## 1, 3, 7, 11; 13, 14,
15; 17; 23, 25.
Hint for ## 3.7.{23,25}: 1) Note that you should not
solve these problems by
substituting
y_C into the DE and collecting like terms!
Instead, the two pieces of the complementary
solution
will allow you to find alpha and beta as explained below.
Note that in
these problems, you are supposed to do
the reverse of what you did in ##
3.7.{1,3,7,11}. Namely, you are
supposed to recognize
the lambda1 and lambda2 from y_C and
use them to find alpha and beta. To do so:
- expand
(lambda -
lambda1)*(lambda - lambda2)
into a quadratic polynomial in lambda;
- associate this quadratic polynomial with y''+alpha*y'+beta*y.
2) The "particular solution" piece of the given y(t) will
allow you to find g(t)
by simply substituting it into
the DE.
Note for #
3.7.25:
What the book refers to as coefficients alpha, beta in the DE
y'' + alpha*y' + beta*y may be confusing, because both the book
and we used
alpha, beta for complex lambda's: lambda = alpha +/-
i*beta.
To avoid this confusion, keep using the latter notation and find
coefficients
b, c in the DE y'' +b*y' + c*y = ...
Answer for # 3.7.14: u1/2 -
u2/8
HW # 17
Sec. 3.8: ## 1, 3, 4, 11, 5 (yes, I recommend doing #5 after
#11), 12; 17; 29,
30; 31.
In the following Word Problems, find y_p:
Word
Problem 1: y'' +
w^2*y = cos(w*t)
Word
Problem 2: y'' +
2*a*y' + w^2*y =
cos(w*t)
Word
Problem 3: y'' +
w^2*y = cos(0.9*w*t)
Use Mathematica to plot all three solutions together for w = 1, a = 0.1, and 0 <= t <= 8Pi.
General
Note:
Use the Table on p. 163 to determine the general form of your y_p.
1) Make sure to carefully read the caption to the Table, which explains
how r has to be chosen. (Note that in only one problem of
those listed
below will you encounter r=2; in all other problems
you will need to
choose between r=0
and r=1.) The concept behind
choosing r was
also presented in Ex. 6(a) of the
Notes and Ex. 6 in the book.
2) Note that the "n" in
"t^n" in the right column matches the "n" in
the left column.
Note for ## 3.8.17: You are not required to find y_p,
but just its form
(see Ex. 6(a) in
the Notes if you are not sure about the difference between the two).
Note for ##
3.8.{29,30}: 1) Begin by
recognizing lambda1 and lambda2,
as done in Example 6 in the Notes.
2) Then find alpha and beta as described in the
Hint for ## 3.7.{23,25} in HW # 16 (i.e., the previous
one).
Also pay attention to what that Hint says you should not do.
Hint for #
3.8.31:
1) Write down the general solution and determine C1, C2
from the condition at t = infinity;
2) Find the particular solution;
3) Find the initial conditions from the now-known y(t).
Answer for # 3.8.4:
e^t*(sin(t) - 2cos(t))/5
Answer for #
3.8.12: (1/2)*(sin(t) - cos(t))
Answer for # 3.8.30: alpha = 0,
beta = 4
Answer for WP
1:
t*sin(w*t)/(2*w)
Answer for WP
2:
sin(w*t)/(2*w*a)
Answer for WP
3:
cos(0.9*w*t)/(0.19*w^2)
HW # 18
Sec. 3.9: ## 1; 4, 5, 7; 16, 17,
18.
Word
Problems: 1. Solve the IVP y'' + w^2*y = sin(w*t),
y(0)=0, y'(0)=0
using the method of Example 3 from Lecture 18.
(Use the final formula in that Example; you do not need to repeat its derivation.
That formula is at the end of Note 1 following Example 3.
Your focus should be on doing the integral in that final formula.
That integral will split into two - see the Note below. Make sure to substitute
the limits in each of them. At the end, they will give two very different terms. )
2. Given the integral I(t) below, find I'(t) and
I'(t0):
(a) I(t) = Integrate[cos(t-s)*g(s),{s,t0,t}];
(b) I(t) = Integrate[e^{t+2s}*g(s),{s,t0,t}];
(c) I(t) = Integrate[ln(t/s)*g(s),{s,t0,t}].
Note for # 3.9.1: Compare the answer with that found by
the method of topic 1b of Lecture 15.
Note for # 3.9.4: To do the integrals for u1 and u2, follow these lines:
1) For the one with e^s in the numerator, use a
substitution e^s = v.
2) For the other, with e^{-s} in the numerator, use this trick:
1/(1+e^s) = e^{-s}/(e^{-s}+1) (why is this true?),
and then use a substitution e^{-s}=v. Then you'll need to do a
simple case
of long division of an expression like v/(v+1).
Look it up in a
Calculus II textbook if you do not remember how to do it.
Note for # 3.9.5: 1) One goal of this problem
is to show you what happens to the term
y1*u1 (see Example 1 in Lecture 18).
2) Compare your y_p with that obtained by the method
of Lecture 17.
(Its form is given in one of the Examples of that Lecture.) By how much
do
they differ and what is the significance of that difference relative to
the DE?
Note for # 3.9.7: Compare your y_p
with that obtained by the method of Lecture 17.
(Its form is similar to that of y_p in Example 5 of Sec.
3.8/book.) By
how much
do they differ
and what is the significance of that difference relative to the DE?
Note for ## 3.9.{16--18}: The notation "lambda" means the
same as "s" in the Lecture Notes.
Note for #
3.9.16: You should find the answer
by comparing the given y_p with that
found in a class example. Then this becomes s straightforward
problem.
Note for # 3.9.{17,18}: Use the
form y = c1*y1 + c2*y2 + y_p,
where you should
view the given y_p as y1*u1 + y2*u2;
from that equation you can deduce y1 and y2.
Note for #
3.9.17: You should use the first
given form of y_p and ignore the form with sinh.
Note for WP: Use the identity sin(a)*sin(b) =
1/2*( cos(a-b) - cos(a+b) ).
Answer for # 3.9.4: 1/2( e^t*(-e^{-t} + ln(1 + e^{-t}) ) -
e^{-t}*ln(1 + e^t) )
Answer for # 3.9.10: t*(ln(t))^3/6
Answer for # 3.9.16: Read an Example in the Lecture
Notes.
Answer for # 3.9.18: alpha = beta = 0, y0 = 0,
y0' = 1.
Answer for WP 1:
G/(2*w)*( sin(w*t)/w - t*cos(w*t) )
Note that, as in
## 3.9.{5,7}, the answers found by the methods of
variation of parameters and undertermined coefficients are different.
How much do they differ by?
Answers for WP 2: (a)
Integrate[-sin(t-s)*g(s),{s,t0,t}] + 1; 1;
(b) Integrate[e^{t+2s}*g(s),{s,t0,t}] + e^{3t}*g(t); e^{3t0}*g(t0);
(c) Integrate[1/t*g(s),{s,t0,t}]; 0.
HW # 19
Sec. 3.10: ## 2, 3, 4, 6; 11(a,c),
12(b,c), 7(modified as
described in the Note below).
Word Problem:
Using Mathematica, plot in
the same figure the
solution of the IVP
y'' + 2a*y' + w^2*y = cos(w*t),
y(0)=0, y'(0) = 0,
for w = 1 and three
values of a: a = 0.10, 0.05, 0.025.
(A command to combine several graphs in one figure is Show; find details on it in Help.)
What will happen to the solution as a is further decreased toward
0?
Notes for ## 3.10.{2,3,4}: 1) Note that the preamble to all these problems (i.e., to 2,
3, 4)
contains the information needed for you to find k (see Lec.
15).
(That is, no numbers in problems 2, 3, 4
themselves are related to k.)
2) You are to start with the
equation of the undamped
oscillator (e.g., use
Eq. (1) in the Notes for Lec. 15, where you set the
damping constant to 0).
Next, add
the force F(t), given in the problems 2,3,4, to the right-hand side.
Then, put your equation in the form of Eq.
(3) of the Notes for Lec. 19
and then use its solution stated in the Notes.
3) Note - again - that the "F" in Eq. (3) in Lecture 19 is not the same as F(t)
given in ## 2,3,4: In those problems, F(t) is the force (i.e.,
function of time)
acting on the oscillator. On the other hand, in Eq. (3), F is the
constant in
front of the cos-term (and after the coefficient of y'' has been made 1
- see
the previous Note).
4)
The initial conditions
are y(0) = y'(0) = 0.
Notes for #
3.10.6: 1)
The
identity referred to in the book should have been:
2*sin(a)*sin(b) = cos(a-b) - cos(a+b).
2) The textbook confusingly puts "m" after the expression for
y(t).
This "m" is not the mass, but the units -- meters.
The mass is denoted by the italic-font letter m.
2) m
can be found from the amplitude of y(t), once you compare with
the formula
found in topic 2 of Lec. 19 or Example 1 in the textbook;
also, recall what Note 2) above for ## 3.10.{2,3,4} said.
3) Then, (k/m)=w0^2
can be found from the frequency
of one of the terms
in the given solution after you use the identity from Note 1) for this
problem.
When you use this identity, recall that cos(-x) = cos(x), i.e.,
cos(a-b)=cos(b-a).
Also, review the formulas above Eq. (8) on p. 177 of the textbook.
Note for #
3.10.11(a): We discussed in class where
each term of the solution comes from.
All that remains for you to do is to verify that the given solution
satisfies the initial conditions.
Note for # 3.10.12:
Accept on faith the solution given by Eq. (12b) on p. 180,
and use that solution as the starting point of your
derivations.
Notes for #
3.10.7: 1) Set k = 128 N/m instead of
the value given in the book.
2) You
may use one of the solutions derived in ## 3.10.{11,12}.
Answer for # 3.10.2(b): (t/10)*sin(10t)
Answer for # 3.10.4(b): (1/9)*sin(t)*sin(9t)
Answer for # 3.10.6: m =
100/(7*pi^2), k = 3600/7.
Answer for #
3.10.7(b): Although the limit of this oscillatory solution
does not exist,
its envelope approaches the
values +/- 5/16.
Extra
credit # 7 (worth
0.2% added to your final
grade)
Assigned: 11/08
Due: 11/15
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 3.10: # 11(b). Hint: Use
L'Hospital's Rule, where you should treat "delta" as the denominator
and the rest of the expression as the numerator.
HW # 20
Sec. 3.11: ## 1, 3; 7, 8, 9,
10; 11, 12, 13, 23, 25; 16, 17, 18;
20, 21.
Sec. 3.13: ## 23, 25.
Hint for # 3.13.23:
See the Note for
##
3.8.{28--30} in HW 17.
Hint for # 3.13.25: Remember
that when handling a nonhomogeneous DE, you must
first handle its homogeneous part, and only then deal with g(t) or y_p.
Thus, to find the coefficients
a,b,c, begin by substituting the
given homogeneous
part of the solution into the homogeneous DE.
However, do not
substitite the
entire expression, c1*y1+c2*y2+c3*y3 !
Rather, substitute one term at a time, and without its constant
cj (j=1,2, or 3).
The justification is as follows. Since c1, c2, c3 are
arbitrary,
you can set them to be as convenient. In particular, you may
take sets {c1, c2, c3} = {1, 0, 0}, {0, 1, 0}, and
{0, 0, 1}.
After you have found
a,b,c, you can easily find g(t).
Answer for # 3.11.8: yes
Answer for # 3.11.10: yes
Answer for # 3.11.12: 1
Answer for # 3.11.16: cosh(t), sinh(t)
Answer for # 3.11.18:
e^{-2t}*(cos(t)+sin(t)), e^{-2t}*sin(t)
Answer for # 3.11.20: A = [
0 1 2
1/2 -1/2 1/2
1/2 1/2 -1/2 ]; yes, a FS
Extra
credit # 8
Assigned: 11/
Due: 11/
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Word Problem 1 (worth
0.15% added to your final
grade):
Prove Abel's theorem for n = 3.
Word Problem 2 (worth
0.15% added to your final
grade):
Prove the contrapositive of Thm. 3.9:
If a set {y1, y2,
... yn} is linearly dependent, then it is not a fundamental set.
Hint: Use Definition-A on p. 20-8 as
well some property/ies of the determinants listed on p. 20-5.
It will suffice if you do the proof for n = 3.
HW # 21
Sec. 3.12: ## 3, 5, 9, 15, 17; 21, 23, 25.
Sec. 3.13: ## 5, 7, 13, 14; 15, 17, 18,
19; 31.
General note for
some of the problems:
You may need the algebraic identities
a^3 - b^3 = (a - b)*(a^2 + ab + b^2)
a^4 - b^4 = (a^2 - b^2)*(a^2 + b^2).
Answer for # 3.13.14: c1 + c2*t + c3*e^{-t} + 2t^2
Answer for # 3.13.18: t^2 * (A1*t*sin(2t) +
A0*sin(2t) + B1*t*sin(2t) + B0*cos(2t) )
Note for # 3.13.31:
As you know, the solution is y_c + y_p, where the two terms
are the
complementary and particular solutions. The complementary solution
has constants c1, c2, etc.. In other problems, you were able to
determine
these constants by using initial conditions. The point of this problem
is to
illustrate that, sometimes, a known asymptotic behavior of the solutuion
(i.e., what it is for t-> +infinity or -infinity) can be used
instead of the
initial conditions to determine (some of) those constants.
HW # 22
Sec. 4.1: ## 1, 2, 4; 7; 16, 17,
19, 21; 25.
Word Problem
1: Let A1,
A2, A3 be columns of a 3x3 matrix A, and let there exist a 3x1 vector x
such that A x = b,
where b is some
3x1 vector.
Prove that b
is a linear combination of the columns of A.
Clarification:
By definition, a linear combination of vectors A1,
A2, A3
is an expression: c1*A1 +
c2*A2 + c3*A3, where c1, c2, c3 are any scalars.
Word Problem
2: Let A1, A2, A3 be columns of
a 3x3
matrix A, and let
there exist a 3x1 nonzero
vector x
such that A x = 0,
where 0 is the 3x1
zero vector.
Prove that the columns of A
are linearly dependent
(i.e. that one of them is a linear combination of the other two).
Hint for # 4.1.25(a): Take A =
[a(t) b; 0 c(t)],
where b = const.
Here and in what follows, the ";" separates the rows of a
matrix or vector.
Hint for #
4.1.25(b): You need to use a formula from p. 219.
Hints for
WPs: 1)
Use the Very Important Property of matrix-vector multiplication.
2) For WP1, the "proof" is really one simple step.
For WP2, the proof is two steps. In the second step, you need to use
the fact that, since vector x
is nonzero, one of its components must be
nonzero andone can
then divide by it.
Answer for # 4.1.2: [2
2t^2+t+2; -4 -2]
Answer for # 4.1.4: t^2*(-t-1)
Answer for # 4.1.16: [-t + c1; t^2 + c2]
Extra
credit # 9
Assigned: 11/6
Due: 11/18
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Problem 1 (worth
0.2% added to your final
grade): ## 4.1.{27, 28} (both
problems must be done for a credit)
Problem 2 (worth
0.2% added to your final
grade): Read Example 1 on pp.
214-215 of the textbook
and do # 4.1.30.
HW # 23
Sec. 4.2: ## 7; 11, 13, 15, 16, 19, 21.
Sec. 4.3: ## 1, 3; 7, 10.
Sec. 4.2 (yes, 4.2 again! - I am just suggesting the order in
which you should do these problems): # 9.
Sec. 4.3: ## 15, 17, 23; 25,
26, 28.
Sec. 3.11: ## 11, 13, 23, 25; 7, 9, 10.
Sec. 4.3: ## 29(b,c), 30(b,c), 31(b,c), 33.
Sec. 4.8: # 11.
Note for # 4.3.23(c): This is just asking you to use the
Very Important Formula from Lec. 22.
Note for # 4.3.23(d): If you
have difficulty finding the inverse of a 3x3 matrix by hand, you
may use Mathematica (type "Matrix Operations" in Help) or Matlab.
Note for ## 3.11.{11,13,23,25}:
First rewrite the given higher-order DE as a system of first-order
DEs
and then apply to it the Liouville formula.
(If you want, you may check your answer by applying the Abel theorem
(Thm. 3.6 in the textbook) directly to the given 2nd- or 3rd-order DE.
However, this should serve only as a check of your work; the main method
that you need to practice is that based on the Liouville formula.
Notes for ## 3.11.{7,9,10}:
1. First, rewrite this as a 1st-order system. Then, consult topic
2c in Lec. 23.
2. The indices "1", "2", etc. of the solutions here have the same
meaning as
in the aforementioned topic 2c. They have a completely different from the
meaning of the indices used in topic 0 of this Lecture.
Hints for ## 4.3.{29,30,31}(b):
1) Since the formula Psi-hat(t) = Psi(t) * C must
hold for any t,
then in order to find the constant matrix C, it will suffice to
do so for one convenient value of t.
2) Equation Psi-hat(t) = Psi(t) * C
implies that
C = (Psi(t))^{-1} *
Psi-hat(t).
3) Use the formula for
the inverse of a 2x2
matrix on p. 217.
You may also both find the inverse and check the result
of your matrix multiplication using Mathematica
(see the note for # 4.3.23 above).
Note for # 4.3.33: While the
solution of this problem is based on a formula from topic 2e
of Lec. 23 and is thus similar to the solution of ## 4.3.{29,30,31}(b),
you can also notice similarity with the solution of ## 4.3.{15,17,23}.
Answer for # 4.2.16: y''' - 4 y'' + 2 y = e^{3t}
Answer for # 4.3.10: yes
Answer for # 4.3.26: W(t) = - e^t / t
Answer for # 4.3.28: -3
Answer for # 3.11.10: yes
Answer for # 4.3.30: (b) C = [2, 1; -1,
3], (c) yes, a FS
Here and in what follows,
the ";" separates the rows
of a
matrix or vector.
Extra
credit # 10
Assigned: 11/11
Due: 11/20
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Problem 1 (worth
0.2% added to your final
grade):
Prove Liouville's formula for n = 2. Follow
these guidelines.
1. Use Fact 4 from p. 20-5.
2. Express the derivative of each entry using the linear system.
3. Follow the proof of Abel's theorem.
Problem 2 (worth
0.25% added to your final
grade):
Liouville's formula (as well as Abel's theorem) can be used as a method
of
Reduction of Order, alternative to that considered in Lecture 13.
Your assignment is to demonstrate this for n = 2 following the steps outlined
below.
1. Consider the linear system (3) of Lecture 23. Let its known solution be y1 = [y11; y21]
(i.e., we know y11
and y21). Our
goal is to find the second solution, y2 = [y12; y22].
2. Write out the DE for either component, y12 or y22. It will have both of
these
components on the r.h.s.. Therefore, we cannot solve it unless we know
a relation between them.
3. Obtain this missing relation from Liouville's formula.
Substitute it into the DE obtained in step 2.
4. Find the solution of the resulting 1st-order DE for the single
component.
5. Write down the form of the second solution y2 = [y12; y22].
HW # 24
Sec. 4.4: ## 29; 13, 15; 19, 20,
23, 27.
Word Problem:
For each matrix given below, find the
eigenvectors.
Do the eigenvectors form a Fundamental Set of solutions?
(a) A = [2, 0; 0, 2]
(b) A = [2, 1; 0, 2]
(c) A = [7, -2, -2; 2, 2, -1; 0, -9, 6];
lambda = 9, 3, 3
(d) A = [5, 2, -2; 1, 4, -1; -3, -3, 6];
lambda = 9, 3, 3
Sec. 4.5: ## 1, 3, 5, 9.
Sec. 4.6: ## 1, 3, 5; 11, 13, 17.
General
note:
You may check some of your answers using commands Eigenvalues and/or Eigenvectors
in Mathematica or eig in
Matlab.
Note for #
4.4.27: The other two eigenvalues are -2 and 1.
Note for # 4.5.9: The eigenvalues
are -3, 1, 2.
(I'd
like to hope that you could obtain them yourself by expanding the
determinant
(A - lambda*I) with respect to the column with most zeros, as you
were probably
taught in your Linear Algebra course.)
Note for ## 4.4.{15,27},
4.5.9, and
Word Problem (c,d):
To find eigenvectors, do not
use Mathematica or Matlab for
anything but checking your work.
(You will be tested on finding eigenvectors of a 3x3 matix, and
you must know how to do so by hand.)
Review your Linear Algebra
notes or textbook on how to solve a 3x3 linear system using
Row-Reduced Echelon Form (a.k.a. Gaussian elimination).
Note that at least one row at the bottom of your Row-Reduced matrix
will be all zeros.
This means that you
will have one or two free/arbitrary variables.
Notes for ## 4.6.{1,3,5}: 1. Recall that when
finding components of an eigenvector of a n x n matrix,
one needs to solve only (n-1) equations.
2. One easy solution to an equation a*x1 + b*x2 = 0
is: x1=-b, x2=a.
Note for #
4.6.17:
You may take the
eigenpairs of A from the answer to # 4.6.1
in the back of
the book.
Answer for # 4.4.20: (b) [2;
-3] and [1; -3]*e^{-3t}; (c) yes
Notes for WP
(a):
1) To do this problem by brute force, obtain equations (if any) on
components
x1, x2 of the eigenvector(s) x, then state the seemingly obvious fact that
x = [x1; x2] and finally follow the related steps of Example 3.
2) Check your answer by comparing it with a fact stated on p. 24-7.
Answer for Word Problem: (a) yes; (b)
no; (c) no; (d) yes
Extra
credit # 11
Assigned: 11/13
Due: 11/22
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Sec. 4.4, # 30 (worth
0.15% added to your final
grade)
Sec. 4.4, # 32 (worth
0.1% added to your final
grade)
Sec. 4.5, # 14 (worth
0.2% added to your final
grade)
HW # 25
Sec. 4.8: ## 29,
31; 15, 16, 17, 18.
General
note:
You may check your answers using command
DSolve in Mathematica
(under Examples there, see Scope and then find a rubric about Linear
Systems;
one of the three related examples shows there how to include
initial conditions in the command).
There are also a couple of technical points that you need will need to know:
1) Unfortunately, Mathematica does not seem to have a syntax to solve a
system of DEs in matrix form. That is, you cannot use a syntax based on
y ' = A y + g, where y, g are vectors and A is a matrix. Instead, you will need
to write the above system as a set of equation and then DSolve them as
examples
in Examples > Scope > Systems of Linear Equations show.
2) While you can easily multiply 2 x 2 matrices and 2 x 1
vectors in the assigned
problems by hand, you may also sometimes want to check their
multiplication with
Mathematica. In that case, keep in mind that the symbol '*' does not denote a
matrix-vector (or matrix-matrix) multiplication in Mathematica.
Instead, the '.' does.
For example, if A = {{a,b},{c,d}} and y = {y1, y2}, then the command to find
A y is A.y .
Note also that in Mathematica, {y1, y2} can stand for both a column and a row.
(In the example above it stands for a column. Mathematica somehow
figures out
what it should be.)
Notes for ##
4.8.{29,31}: 1) Solving an IVP where the initial condition is a matrix
uses the same idea as the IVP with a vector initial condition.
(We solved IVPs with vector IC in Lec. 24.) To relate
vector IC to matrix ones, use one of the properties on
p. 22-6 of posted Notes.
Alternatively, see Thm. 4.5 (Lec. 23) or Example 2 in Sec. 4.8.
In the notations of Thm. 4.5, Psi(t) is the
fundamental matrix
that you find first, and Psi-hat is the matrix whose value
at t=t0
is given as the initial condition.
2) Ex. 7 of Lec. 24 may be useful for one of these problems.
Extra
credit # 12 (worth
0.15% added to your final
grade)
Assigned: 11/15
Due: 12/02
Note: Before
you attempt this
extra-credit assignment, please read these
instructions!
Word Problem:
Consider a nonhomogeneous scalar DE and
rewrite it as a system of two first-order DEs.
Apply the Method of Variation of Parameters to this system and show
that the resulting
equations for u1 and u2 (where u = [u1; u2]) are the same as those in Sec.
3.9.
HW # 26
Sec. 5.1: ## 1, 2, 3, 4; 9; 16, 17, 19, 21; 32, 33, 35;
23, 37, 39.
General note:
You may check your answers involving integration using command Integrate in Mathematica.
To integrate within infinite limits, use {t,0,Infinity}; to integrate
in finite limits, use {t,0,T}.
Hint for # 5.1.9:
Write t = (t - 1) + 1 in the exponent and use
a new variable z = (t - 1) for the integration.
Hint for # 5.1.19: Write t = (t - 2) +
2 in the exponent and use a new variable
z = (t - 2) for the integration.
(The point in your doing this problem is to demonstrate that the
answer will be different from that for a similar-looking
function h(t-2)*sin(w(t-2)), where you'll learn the meaning
of h(t-2) in Theorem 5.4. in Lec. 27.)
Hint for # 5.1.21: Denote (s+2) by a new
letter and notice close similarlty with # 16.
The more general case will be considered in Theorem 5.4 in Lec. 27.
Note for ## 5.1.{32,33,35}: You did all of these
integrals in Calculus I or II.
HW # 27
Sec. 5.2: ## 1, 2, 3, 6, 5, 4, 10, 9, 11;
25, 23; 13, 14, 15, 16, 17, 18; 39, 41;
42 [but assume g(t)=12 for all
t>0], 43, 44, 45.
In addition,
for ## 5.2.{15,17,18}, sketch f(t).
For that, first sketch the
non-shifted function
(i.e., the inverse Laplace transform of the given F(s) without the exponential factor),
and then shift your graph as prescribed by the appropriate Shift theorem.
Hint for # 5.2.9:
t = (t - 2) + 2
Hint for # 5.2.11: You can
either apply one of the shift theorems directly
or use the above Hint for # 5.2.9.
Note for # 5.2.39: Find the partial
fraction expansion by hand.
Note for # 5.2.41: Review the class
Examples 2(b) and 2(c) in comparison with one another;
also review the must-read Example 2 in the textbook.
Note for ## 5.2.{44,45}: You may use command Apart in
Mathematica to find the
partial fraction expansion.
(However, for ## 42, 43 you need to do partial fraction expansion by
hand!)
Hint for # 5.2.42: g(t) is
proportional to the function shown in Fig. (a) at the top of p. 342.
Note for # 5.2.44: You may obtain
the answer using DSolve in Mathematica.
Answer for # 5.2.14: 2 sin(5t) + 4e^{3t}
Answer for # 5.2.16: (5/6)*t^3*e^{3t}
Answer for # 5.2.18: h(t-2) * e^{9(t-2)}
Answer for # 5.2.42: 3( 1 - e^{-4t} ) + 2 e^{-4t}.