Math 230.A  / Fall 2022

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:   sec1.2sec1.3

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)?
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 #1.2.14:   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--2.4Answers 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.
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 list 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,

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) A formula for the solution of the general case of the Cooling Law
(but still with
S(t) = S0), found on p. 36, will be helpful.
Note for Sec. 2.9:    Read pp. 78, 79 (Drag force, Case 1).

Answers for # 2.2.4,14,20,28,36:    See the Note above.

Extra credit # 1
Assigned:  09/07
Due:   09/14
Word Problem 1:
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?
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:
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. Jim processed 25 units of product during
the first hour and 45 units during the second hour. 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  CC*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), 14, 15.
(Pages from the textbook that you will need are posted in HW # 2.)
Sec. 2.3:  ##  16, 17;   2, 3, 1;   7(a,c), 6.
Sec. 2.9:  ##  4, 1.

General Note:    One can apply the solution found in topic 5 of Lecture 2 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.
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.)

In Lecture 2, I recommended that you use the method of its topic 5
(found on page 2-11 there) to solve such equations.
Note for ## 2.3.{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. (Incidentally, the solution
found there can be obtained by the method of topic 5 of Lec. 2.)

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.

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.
Note for Sec. 2.9:            Read pp. 78, 79 (Drag force, Case 1).
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 #2.3.6:           (a) 400 min,  (b) 7.5 lb,   (c) 10 lb at 200 min
Answer for #2.9.1:           Using the solution at the top of p. 79, we obtain: T = (m/k)*ln(2).
Answer for #2.9.4:           t = ln( (v0+mg/k) / (mg/k) ) *(m/k)

Assigned:  09/14
Due:    09/21
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 Example 3 of Sec. 2.9. Apply the technique of p. 82
and Example 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.

"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  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.)
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 # 2.6.24:             tan(pi/4 + t) - 1
Answer for # 2.5.12:             (a)  2;   (b)  0.

Assigned:  9/16
(source:  "Fundamental of Differential Equations" by R.K. Nagle, E.B. Saff, A.D. Snider; Chap. 3)
- 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).)
(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)

"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
## 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,
u
se 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  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.

Assigned:  09/21
Due:   10/01
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 # 2.8.4:    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''
at that value of  t  where you know all other terms in the DE.
(I'm being vague here because I want you to think 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
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.

Assigned:  10/02
Due:  10/09
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.
In addition, explain why the graph
is positive on one particular side of
t
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
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
t (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  t
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
Answer for # 3.3.18:     (a)=C, (b)=B.
Answer for # 3.3.20:     (c)  x0 + (m/k)*v0

Assigned:  10/11
Due: 10/18
Sec. 3.3:   # 16.

HW # 13
Sec. 3.4:  ##  1, 3, 5, 6, 7, 10;   11, 12;   13.
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.
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.

Assigned:  10/13
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).
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).

(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
"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.{3,7,9,11}:
olving 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.
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.
Note for # 3.5.24,26:     To check your answers, 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
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
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)

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.)

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)
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.

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.)

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.
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).
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

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 # 3.9.18:    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  aa = 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 # 3.10.6:       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.

Assigned:  11/08
Due:  11/15
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 # 3.11.16:    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/
Word Problem 1
Prove Abel's theorem for n = 3.
Word Problem 2
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 # 4.1.16:  [-t + c1;  t^2 + c2]

Extra credit # 9
Assigned:  11/27
Due:  12/04
Problem 1
(worth 0.2% added to your final grade):    ## 4.1.{27, 28} (both problems must be done for a credit)
Problem 2
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.
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 # 4.3.26:    W(t) = - e^t / t

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/
Due:  12/
Problem 1

Prove Liouville's formula for  n = 2Follow 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
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:  12/
Due:  12/
Sec. 4.4, # 30

HW # 25
Sec. 4.8:  ##   29, 31;   15, 16, 17, 18.

General note:
(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.

Assigned:  12/08
Due:  day of final exam
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}