y'''-8y=e^ix by the method of variation of parameters

3) Solve the initial value problems

c) R′ + (R/t) = (2/(1+t² )) , R(1) = ln 8.

e) ) cos θv′ + v = 3 , v(π/2) = 1.

5) Express the general solution of the equation x ′ = 2tx + 1 in terms of the erf function.

7) Solve x ′′ + x ′ = 3t by substituting y = x ′

9) Find the general solution to the differential equation x ′ = ax + b, where a and b are constants, first by separation of variables, and second by integrating factors

Find a system of recurrence relations for the number of n-digit quaternary sequences that contain an even number of 2’s and an odd number of 3’s. Define the initial conditions for the system. (A quaternary digit is either a 0, 1, 2 or 3)

Let (x_n), (y_n) be bounded sequences.

a) Prove that lim inf x_n + lim inf y_n ≤ lim inf(x_n + y_n) ≤ lim sup(x_n + y_n) ≤ lim sup x_n + lim sup y_n. Give example where all inequalities are strict.

b)Let (z_n) be the sequence defined recursively by z₁ = z₂ = 1, z_n+2 = √ z_n+1 + √ z_n, n = 1, 2, . . . . Prove that (z_n) is convergent and find its limit. Hint; argue that 0 < z_n < 4 for all n and use part (a).

If V = U ⊕ U^⟂ and V = W ⊕ W^⟂, and if S₁: U → W and S₂: U^⟂ → W^⟂ are isometries, then the linear operator defined for u₁ ∈ U and u₂ ∈ U^⟂ by the formula S(u₁ + u₂) = S₁u₁ + S₂u₂ is a well-defined linear isometry. Prove this.

What is the optimal number to order to maximize Joseph's expected profit and what is the expected profit on the following?

Joseph must order each year's calendar in August for delivery and eventual sale by December 31st. In previous years, standard Business Calendar sales have ranged from a low of 2000 to a high of 2300, so assume that demand is uniformly distributed between 2000 and 2300. Each calendar costs $17.90 and sells for $25.96, and unsold calendars usually sell out at a 1/2 price sale in January. Assume Joseph can only order in quantities of 50.

Is Z₇^x a cyclic group? If so, find all the generators. Same for Z^x₈, Z₁₀^x, Z^x₁₁, Z^x₁₂

a) Suppose that a ∈ Z is a unit modulo n. Prove that its inverse modulo n is well defined as a residue class in Zn, and depends only on the residue class a in Zn.

b) Let Z × n ⊆ Zn be the set of invertible residue classes modulo n. Prove that Z × n forms a group under multiplication. Is this group a subgroup of Zn?

c) List the elements of Z × 9 . How many are there? For each residue class u ∈ Z9, compute the elements of the sequence u, u2 , u3 , u4 , . . . until the pattern is clear. Determine the length of each repeating cycle. Is Z × 9 a cyclic group?

Concepts: Transformations of functions and finding intercepts.

HMS Nautical Inc.

2200 Seaview Blvd.

North Shore, Hi 09231

Math Class

DeVry University

122^nd and Pecos

Westminster, CO 80241

Dear Students:

I have recently been employed by HMS Nautical Inc to work on their submarine program. I have only some basic data to work with and no idea how to use it to get the information I need.

Here is what I know. First, I know that our Subs have a maximum running depth of 500 feet below sea level. I also know that a sub functioning at acceptable levels should be able to reach maximum depth in 10 minutes. Finally I know that I need to multiply the decent by a factor of 5 to achieve an accurate model. I also have a chart that lists times and depths for the sub.

Finally, I know that the sub follows a quadratic model when it descends and then ascends.

I have been told that you will be able to take this data and make sense of it. I would like a model for the path of the submarine as it descends to its running depth and then returns to the surface. I want to be able to use this model to predict where the sub will be at any time during its decent/ascent cycle. I would also like to know after how many minutes I should expect the sub to breech the surface of the water again.

Please explain clearly how you came up with the model so that I can repeat the process for new additions to our fleet of submarines. I appreciate any help you can give me in this matter. I would like your response returned to me either as a business letter or a narrated PowerPoint presentation.

Sincerely,

Nemo Hook

lets say you are intetested in how levels of trade for countries influence economic growth. How would you structure the dataset to analyze this relationship

your project status end of 10th day.Discuss the status of this project in terms of schedule and budget using EVM.What is your estimate budget at completion if your performance continuous as is planned?Based on the current performance update the expected completion time,and how efficiently must we work from on to complete the project within budget and schedule?

Prove that cardinality is an equivalence relation. Prove for all properties (refextivity, transitivity and symmetry). Please do this problem and explain every step. The less confusing notation the better, thanks!

You owe $1500 on a credit card that charges 2% interest each month. You can pay $50 each month with no new charges. What is the equilibrium value? When will the account be paid off?

A) Equilibrium value is 2500 and the account will be paid off approximately after 47 months.

B) Equilibrium value is 50 and the account will be paid off approximately after 47 months.

C) Equilibrium value is 2500 and the account will be paid off approximately after 40 months.

D) Equilibrium value is 1500 and the account will be paid off approximately after 40 months.

E) All the above is incorrect