In parts 1–2 below, determine whether or not ? is a subgroup of ?. (Assume that the operation of ? is the same as that of ?.). (2)

1. ? = 〈R, +〉, ? = {log?: ?∈Q, ?>0}.

2. ? = 〈R, +〉, H = {log?: ?∈Z, ?>0}.

Was the Elements an exposition of the most advanced mathematics of its time? (Elements by Euclid)

Find the solution of the initial value problem y′′−2y′−3 y=15te2t, y(0)=2, y′(0)=0.

Obtain the general solution of the following equations:

r Utt − c^2 r Urr − 2 c^2 Ur = 0, c = constant,

could you explain the differences and the similarities between undetermined coefficients method and annihilator method? also, could you please solve an example step by step to explain how the annihilator method works?

Exercise (a)

Use Euler's method with each of the following step sizes to estimate the value of y(1.6), where y is the solution of the initial-value problem y' = y, y(0) = 6.

(i)    h = 1.6

(ii)    h = 0.8

(iii)    h = 0.4

Exercise (b)

We know that the exact solution of the initial-value problem in part (a) is y = 6e^x. Draw, as accurately as you can, the graph of y = 6e^x, 0 ≤ x ≤ 1.6, together with the Euler approximations using the step sizes in part (a). (Your sketches should resemble the figures for the first Euler approximation, Euler approximation with step size 0.5, and Euler approximation with step size 0.25.) Use your sketches to decide whether your estimates in part (a) are underestimates or overestimates.

Exercise (c)

The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value of y(1.6), namely, 6e^1.6. What happens to the error each time the step size is halved?

Subject is differential equations.

Find the first four nonzero terms in a power series expansion about x=0 for a general solution to the given differential equations.

a) y' + (x+5)y = 0

b) y'' + (x+5)y' + y = 0

c) y'' + 4xy' - y = 0; y(0) = 8, y'(0) = 0

d) y'' + (x-6)y' - y = 0; y(0) = -5, y'(0) = 0

Please show work! Thank you

What conditions have to be checked to determine if a function is "well defined"? How would I show that a function is not well defined?

A steel company is considering the relocation of one of its manufacturing plants. The company’s executives have selected four areas that they believe are suitable locations. However, they want to determine if the average wages are significantly different in any of the locations, since this could have a major impact on the cost of production. A survey of hourly wages of similar workers in each of the four areas is performed with the following results. Do the data indicate a significant difference among the average hourly wages in the three areas? Hourly Wages ($) Area 1 Area 2 Area 3 21 10 15 21 24 11 24 18 18 17 22 14 15 22 15 18 16 14 11 24 12 22 21 16 Step 1 of 2 : Find the value of the test statistic to test for a difference in the areas. Round your answer to two decimal places, if necessary.

6. This question is about the sequential moves Stag Hunt game. There are two players. Player 1 moves first, player 2 observes player 1’s move, and then player 2 moves. Players get 10 jollies each if they both choose Stag; 5 jollies if they choose Hare; and 0 jollies if they choose Stag but the other player does not.

   1. What is the set of (pure) strategies for player 1? What is the set of (pure) strategies for player 2?

   2. Explain why player 2 has more pure strategies than player 1.

   3. What is the set of SPNE in this game? What is the outcome of the game in this SPNE?

   4. Why are the sets of SPNE in the extensive form, and of NE in the simultaneous move game in strategic form, different?

   5. If the game were changed so player 2 moved first, would the SPNE outcome be different?

Consider the differential equation

2y^2+10y+12 = t+e^t

Find the complementary function and particular integral. Hence write down the full general solution

Solve the given system of differential equations. ??/?? = ? + 4? ??/?? = ? + y

Exercise 1.13.5: Determine and prove whether an argument in English is valid or invalid.

Prove whether each argument is valid or invalid. First find the form of the argument by defining predicates and expressing the hypotheses and the conclusion using the predicates. If the argument is valid, then use the rules of inference to prove that the form is valid. If the argument is invalid, give values for the predicates you defined for a small domain that demonstrate the argument is invalid.

QUESTION A AND B ALREADY SOLVED. PLEASE solve part C, D, E using same method, and give some explantion for your answer. THANKS!

The domain for each problem is the set of students in a class.

(a)

Every student on the honor roll received an A.
No student who got a detention received an A.
No student who got a detention is on the honor roll.

• H(x): x is on the honor roll
• A(x): x received an A.
• D(x): x got a detention.

∀x (H(x) → A(x))
¬∃x (D(x) ∧ A(x))
∴ ¬∃x (D(x) ∧ H(x))

Valid.

(b)

No student who got an A missed class.
No student who got a detention received an A.
No student who got a detention missed class.

• M(x): x missed class
• A(x): x received an A.
• D(x): x got a detention.

¬∃x (A(x) ∧ M(x))
¬∃x (D(x) ∧ A(x))
∴ ¬∃x (D(x) ∧ M(x))

The argument is not valid. Consider a class that consists of a single student named Frank. If M(Frank) = D(Frank) = T and A(Frank) = F, then the hypotheses are all true and the conclusion is false. In other words, Frank got a detention, missed class, and did not get an A.

(c)

Every student who missed class got a detention.
Penelope is a student in the class.
Penelope got a detention.
Penelope missed class.

(d)

Every student who missed class got a detention.
Penelope is a student in the class.
Penelope did not miss class.
Penelope did not get a detention.

(e)

Every student who missed class or got a detention did not get an A.
Penelope is a student in the class.
Penelope got an A.
Penelope did not get a detention.

Two tanks contain a mixture of water and alcohol with tank A contain- ing 500 L and tank B 1000L. Initially, the concentration of alcohol in Tank A is 0% and that of tank B is 80%. Solution leaves tank A into B at a rate of 15 liter/min and the solution in tank B returns to A at a rate of 5 L/min while well mixed solution also leaves the system at 10 liter/min through an outlet. A mixture of water and alcohol enters tank A at the rate of 10liter/min with the concentration of 10% through an inlet. What will be the concentration of the alcohol of the solution in each tank after 10 mins?