Please show all steps and write neatly. Make sure the proving is with letters and not with numbers. Please!!!! Thank you

Let A, B, C be 3 x 3 matrices. Show (AB)C= A(BC)

(From 4.2) In a mass-spring system, motion is assumed to occur only in the vertical direction. That is, the system has one degree of freedom. When the mass is pulled downward and then released, the system will oscillate. If the system is undamped, meaning that there are no forces to slow or stop the oscillation, then the system will oscillate indefinitely. Applying Newton’s Second Law of Motion to the mass yield the second-order differential equation ? ′′ + ? 2? = 0 where ? is the displacement at time ? and ? is a fixed constant called the natural frequency of the system.

a) Verify that the general solution of the above differential equation is ?(?) = ?1????? + ?2????? where ?1 and ?2 are arbitrary constants.

b) Show that the set of all functions ?(?) form a vector space.

solve
cos^2xsinx dy +(ycos^2x-1)dx=0

Casey Nelson is a divisional manager for Pigeon Company. His annual pay raises are largely determined by his division’s return on investment (ROI), which has been above 24% each of the last three years. Casey is considering a capital budgeting project that would require a $4,450,000 investment in equipment with a useful life of five years and no salvage value. Pigeon Company’s discount rate is 20%. The project would provide net operating income each year for five years as follows: Sales $ 4,300,000 Variable expenses 1,960,000 Contribution margin 2,340,000 Fixed expenses: Advertising, salaries, and other fixed out-of-pocket costs $ 790,000 Depreciation 890,000 Total fixed expenses 1,680,000 Net operating income $ 660,000 Click here to view Exhibit 13B-1 and Exhibit 13B-2, to determine the appropriate discount factor(s) using tables. Required: 1. What is the project’s net present value? 2. What is the project’s internal rate of return to the nearest whole percent? 3. What is the project’s simple rate of return? 4-a. Would the company want Casey to pursue this investment opportunity? 4-b. Would Casey be inclined to pursue this investment opportunity?

How would you control for these person-specific and time-specific effects in a panel data regression? d) Can the regression be used to estimate the effect of gender on an individual’s earnings? Explain. e) Can the regression be used to estimate the effect of the national unemployment rate on an individual’s earnings? Explain. f) In the context of the regression you suggested for part (c), explain why the regression error for a given individual might be serially correlated. g) In the context of the regression you suggested for part (c), explain why the regression error across individuals for a given point in time might be correlated.

Find Aut(Z ₁₅). Use the Fundamental Theorem of Abelian Groups to express this group as an external direct product of cyclic groups of prime power order. (Thank you. If any theorems. corollaries, definitions, etc. are related, I'd appreciate the extra knowledge. If you don't have time, I understand and thanks, anyway.

The software application development company for which you work is evaluating two different options for starting a new application development within the upcoming year. The company seeks to earn a return on each project that exceeds its cost of capital of 6.5 %. Further, the company assumes that given the current economic conditions, inflation may be expected to grow by 1% for the next year. Assume that the project takes place in Year 0 and begins earning cash flows on its application product from years 1 through 5.

Answer the following questions based on the analysis of the given cash flows for Project A versus Project B:

What is the NPV for each proposed project?

Which project would you select based on the NPV?

Alter the discount rate based on the method presented in the videos in the Financial Analysis Tools section below so that you reach an NPV of "0" (or close to "0") thereby arriving at the project IRR. Which IRR is higher?

If inflation were not a factor, would your results change?

Write down your own verbal description for a dynamical system, and write down the equations for the dynamical system. (It is fine if this system is very simple! It is also fine if the scenario is not very realistic, as long as the equations match the description!)   

Solve the given differential equation by undetermined coefficients (superposition approach)

y'' + 2y' − 3y = (x^2 + x + 1) + e^−3x

Question 1. A soil management specialist was studying the relationship between the average temperature (in ºC) and the yield in bushels per acre for a certain crop. The data is given in Table 1 below.

Table 1

1. Construct a scatter a scatter diagram for the paired data.
2. As temperature increases, does yield appear to increase, decrease, or remain the same?
3. Compute the coefficient of correlation, and interpret it.
4. Compute the coefficient of determination and interpret it.

The “standard” bagel has always been the one obtained by rotating the circle of radius 3/4 inch centered at (3/4 , 0) around the line x = −1.

Recently, a Californian chef created a bagel with the same circle but rotated around the line x = −2.

(a) Physically, what’s the difference between the two bagels?

(b) What are the volumes of the standard and Californian bagels?

Given is a population of wolves (W) and rabbits (R). R[t+1] = R[t]+ g*R[t] * (1 – R[t]/K) - sR[t]W[t] W[t+1] = (1-u)W[t] + vR[t]W[t] Where the carrying capacity of rabbits is 1 million. The growth rate of rabbits is 10% a year and s is equal to 0.00001, v is 0.0000001, and u is equal to 0.01. How many wolves and how many rabbits exist in the equilibrium?

Show that the worst-case number of entries computed by the refined dynamic programming algorithm for the 0-1 Knapsack problem is in Ω(2n). Do this by considering the instance in which W = 2n −2 and wi = 2i−1 for 1 ≤ i ≤ n.

Please answer with clear explanations!!!

Excerpt From: Richard Neapolitan. “Foundations of Algorithms.”