Consider the second-order boundary value problem

y′′ +(2x^2 +3)y′ −y =6x, 0≤x ≤1, (4)

y(0) = 1, y(1) = 0.
(a) Rewrite the second-order equation (4) as a system of two first-order equations

involving variables y and z. [2]

(b) Suppose that yn and zn are approximations to y(xn) and z(xn), respectively, where xn = nh, n = 0,...,N and h = 1/N for some positive integer N. Find the iterative formula when using the modified Euler method to approximate (4) with the modified boundary conditions:

y(0) = 1, y′(0) = z0.
(c) Hence, employ the shooting method, with underlying modified Euler method, to

find approximations yn, n = 1, . . . , N to problem (4)-(5), when N = 5. [Hint: Notice that differential equation (4) is linear.]

Boris and Natasha agree to play the following game. They will flip a (fair) coin 5 times in a row. They will compute S = (number of heads H – number of tails T).

a) Boris will pay Natasha S. Graph Natasha’s payoff as a function of S. What is the expected value of S?

b) How much should Natasha be willing to pay Boris to play this game? After paying this amount, what is her best case and worst case outcome?

This time, after 5 flips of the coin, if there are more heads H than tails T, Boris will pay Natasha H – T. If there are more tails T than heads H, Boris will pay Natasha nothing.

c) Graph Natasha’s payoff as a function of S = H – T. What does this graph remind you of?

d) What is the expected value of Natasha’s payoff? How much should she be willing to pay to play this game? After paying this amount, what is her best case and worst case outcome?

3. For the inhomogeneous differential equation x ′′ + 2x ′ + 10x = 100 cos(4t),

(a) Describe a system for which this differential equation would be an appropriate model.

(b) Find the general solution, x(t), to the equation.

(c) Does the general solution have the expected terms? What behavior do the terms describe?

(d) Find the specific solution that fits the initial conditions x(0) = 0 and x ′ (0) = 0.

(e) Plot the solution and discuss how you see the expected behaviors

This is a combinatorics problem

Suppose we wish to find the number of integer solutions to the equation below, where 3 ≤ x1 ≤ 9, 0 ≤ x2 ≤ 8, and
7 ≤ x3 ≤ 17.

x1 + x2 + x3 = r

Write a generating function for this problem, and use it to solve this problem for r = 20.

Find Laplace transform of

50x''+ 100x' + 60x = 0, x'(0)=0, x(0)=0

Part I On a certain university campus there is an infestation of Norway rats. It is estimated that the number of rats on campus will follow a logistic model of the form P(t)=50001+Be−ktP(t)=50001+Be−kt.

A) It is estimated that there were 500 rats on campus on January 1, 2010 and 750 on April 1, 2010. Using this information, find an explicit formula for P(t)P(t) where tt is years since January 1, 2010. (Assume April 1, 2010 is t=.25t=.25.)
P(t)= P(t)=  .

B) What was the rat population on October 1, 2010?
rats.

C) How fast was the rat population growing on April 1, 2010?
rats per year.

D) According to our logistic model, when will the rat population hit 2,500 rats?
years after January 1, 2010.

E) Rats live in communal nests and the more rats there are, the closer they live together. Suppose the total volume of the rats' nests is F=0.64P+4−−−−−−−−√−2F=0.64P+4−2 cubic meters when there are PP rats on campus.
When there are 750 rats, what is the total volume of the rats' nests and how fast is the mass of nests growing with respect to time?
The total volume is  cubic meters and the volume is increasing at  cubic meters per year.

F) One of the reasons that the rats' population growth slows down is overcrowding. What is the population density of the rats' nests when there are 750 rats and how fast is the population density increasing at that time?
The population density is  rats per cubic meter and the population density is increasing at  rats per cubic meter per year.

Assume that the matrix A is row equivalent to B. Without​ calculations, list rank A and dim Nul Upper A. Then find bases for Col​ A, Row​ A, and Nul A.
A= [1,1,-2,0,1,-3;1,2,-3,0,0,-6;1,-1,0,0,1,7;1,4,-4,1,13,-11;1,4,-5,0,3,-32] B=[1,1,-2,0,1,-3;0,1,-1,0,-1,-3;0,0,1,1,15,1;0,0,0,0,1,-2;0,0,0,0,0,1]

3. A screening test is applied to a population of 1,000 in which the prevalence of the disease in question is 10%. The sensitivity of this test is 96% and the specificity is 92%. The cost of screening the 1000 people is $50 (fifty dollars) per person. A newer screening test has the same sensitivity, a specificity of 96%, but costs $0.50 (fifty cents) more per test than the older screening test. For each person with a positive result from either test, a follow-up diagnostic test must be done that costs an additional $50 (fifty dollars).
   1. Calculate how much money would be saved or lost by choosing the newer screening test (1)?

Provide two (2) examples of problems where the output response(s) is in the form of a vector as opposed to a scalar. Describe each one briefly.

Let gcd(a, p) = 1 with p a prime. Show that if a has at least one square root, then a has exactly 2 roots. [hint: look at generators or use x^2 = y^2 (mod p) and use the fact that ab = 0 (mod p) the one of a or b must be 0(why?) ]

Define ?? [?] = {?(?) ∈ ?[?]|deg ?(?) ≤ ?}. One can show ?? [?] is a vector space. Let ? = {?(?) ∈ ℝ4 [?]|?(6) = 0}.

a) Find a basis for ?. Assume that for any field ?, ?? [?] is a vector space

b) Extend the basis in part (a) to a basis of ℝ4 [?].

Consider the group G = {1, −1, i, −i, j, −j, k, −k} under multiplication. Here
i2= j2= k2= ijk = −1. determine which of the following sets is a subgroup
of G. If a set is not a subgroup, give one reason why it is not.
(a) {1, −1}

(b) {i, −i, j, −j}

(c) {1, −1, i, −i}

(d) {1, i, −i, j}

(Lot sizing) The demand for a product is known to be dt units in periods t = 1,...,n. If we produce the product in period t, we incur a machine setup cost ft which does not depend on the number of units produced plus a production cost pt per unit produced. We may produce any number of units in any period. Any inventory carried over from period t to period t + 1 incurs an inventory cost it per unit carried over. Initial inventory is s0. Formulate a mixed integer linear program in order to meet the demand over the n periods while minimizing overall costs.

1. Consider the implication: If it is snowing, then I will go cross country skiing. (a) Write the converse of the implication. (b) Write the contrapositive of the implication. (c) Write the inverse of the implication.

2. For each of the following, use truth tables to determine whether or not the two given statements are logically equivalent using truth tables. Be sure to state your conclusions. (a) p → (q ∧ r) and (p → q) ∧ (p → r) (b) (¬p ∧ (p → q)) → ¬q and T

3. Give a two-column proof in the style of Section 2.6 which shows the following symbolic argument is valid. (¬p ∨ q) → r s ∨ ¬q ¬t p → t (¬p ∧ r) → ¬s ∴ ¬q 4. Let K(x, y) be the predicate ”x knows y” where the domain of discourse for x and y is the set of all people. Use quantifiers to express each of the following statements. (a) Alice knows everyone. (b) Someone knows George. (c) There is someone who nobody knows. (d) There is someone who knows no one. (e) Everyone knows someone. 5. Again consider the predicate K(x, y) defined in Exercise 4. Negate in symbols the propositions (a), (c), and (e) from Exercise 4. Note: Of course, an easy way to do tis is to simply write ¬ in front of the answers for Exercise

4. Don’t do that! Give the negation with no quantifiers coming after a negation symbol.

6. One more time, consider the predicate K(x, y) from Exercises 4 and 5. Negate in smooth English the propositions (a), (c), and (e) from Exercise 4. Note: An easy way to do this is to simply write It is not the case that ... in front of each proposition. Don’t do that! Give the negation as a reasonably natural smooth English sentence.

7. Three mathematicians are seated in a restaurant. The server: ”Does everyone want coffee?” The first mathematician: ”I do not know.” The second mathematician: ”I do not know.” The third mathematician: ”No, not everyone wants coffee.” The server comes back and gives coffee to the mathematicians who want it. Which mathematicians received coffee? How did the waiter deduce who wanted coffee?

A circus clown is to be fired from a cannon at an angle of 60 degress to the horizontal with an initial   

      speed of 60 ft/sec. The clown is supposed to hit a cushion located 100 ft away. The circus is

      held in a building with 50 ft ceilings.

1. Can the performer be fired to the cushion without hitting the ceiling?
2. Will the clown land on the cushion? If not, will he land short or long of the mark?
3. What angle should be used to have the clown land directly on the cushion?