Using Variation of Parameters: (Higher Order DE) Find the general solution of y'''− 2y''− y'+ 2y = e^x?

1. For each of the following indexed collections of sets {Ai : i ∈ I}, determine ∪i∈I Ai, ∩i∈I Ai, and whether or not the collection is pairwise disjoint. Prove your claims.

   (a) I is the set of prime numbers and for each p∈I,Ap ={n∈N: p|n}.
   (b) I=R≥0 ={r∈R:r≥0} and for each r∈R≥0,Ar ={(x,y)∈R^2 : sqrt(x^2+y^2) =r}

Given f(x) = ax² + k (where f(x) is a quadratic), describe when its reciprocal function would and would not have vertical asymptotes. Refer to the values (+,-,0) of a and k in your description. What are all the possible scenarios?

A company is planning a plant expansion. They can build a large or small plant. The payoffs for the plant depend on the level of consumer demand for the company's products. For the large plant, the company expects $90 million in revenue if demand is high and $40 million if demand is low. For the small plant, the company expects $55 million in revenue if demand is high and $20 million if demand is low. The cost of the large plant is $5 million. The small plant cost is $1 million.

The company believes that there is a 72% chance that demand for their products will be high and a 28% chance that it will be low.

Construct a payoff and regret matrix based on the given information.

What is the decision according to the EMV criterion? Be sure to support your answer.

Is your decision sensitive to the demand assumptions given for this problem? You may answer generally, such as “very sensitive” or “not very sensitive” as long as you support your answer.

The company can pay a market research firm to survey consumer attitudes towards the company's products. The market research firm cost is $100,000. The market research firm has provided the following cross tabulation showing recent results.

Construct a decision tree for this problem showing when the survey is conducted but the plant has not been built. Include all relevant probabilities on the tree and all EMVs needed.

What is the expected value of the market research survey data?

Should the market research firm be hired at a cost of $100,000? Be sure to support your answer.

verify that the function det has the following properties

1. det(... ,v,..., v,...)=0

2. det(...,cv+dw,...)=c*det(...,v,...)+d*det(...,w,...)

3. use the above properties and normalization to prove that

A= det(...,v+cw,...w,...)=det(...,v,...,w,...)

B=det(...,v,...,w,...)= (-1)*det(...,w,...,v,...) and

C= det [diag (λ1, ... , λn ) ] = Πiλi

Just (c) from the problem below:

1.11. Let f be a one-to-one smooth map of the real line to itself. One-to-one means that if f(xi) = f(x2), then x1 = x2. A function f is called increasing if x1 〈 x2 implies

f(x1 )くf(x2), and decreasing if x1 〈 x2 implies f(x1 ) 〉 f(x2 )

Matlab code for Gauss Siedel with solved example in Matlab

Write the double-mass-spring equations as a 1st-order system in matrix-vector form.

show all workings please. TIA

1) a) Write down the augmented matrix for the following system of linear equations and solve the system.

3x₁ − 7x₂ + 4x₃ = 10

−x₁ − 2x₂ + 3x₃ = 1

x₁ + x₂ + 2x₃ = 8

b) Jason invested $30 000, splitting it between the three companies Acorn Industries, Balderdash Bank and Chester Challenge. The interest rates were respectively 5%, 6% and 7% per annum. The total annual income (i.e. the sum of the interest payments) was $1700. The income from the investments with Balderdash and Chester was $200 more than the income from the Acorn investment.

Begin by letting the investments with Acorn, Balderdash and Chester be $a, $b and $c respectively

i) Write equations representing three facts about the investments. Discuss your equation as to why it should be this.

ii) Solve this system of equations using an augmented matrix.

iii) How much did Jason invest with each company? Write an answer in words. (Answers to word problems should always be given in words).

A group of retailers will buy 84 televisions from a wholesaler if the price is $400 and 124 if the price is $350. The wholesaler is willing to supply 76 if the price is $315 and 156 if the price is $405. Assuming the resulting supply and demand functions are linear, find the equilibrium point for the market.

(q, p) = ( , )

Prove and give a counterexample: If N ⊲ G and G ⊲ H, then N ⊲ H.

Please write an explanation with some details

exapnd xy + siny upto third degree using Taylor series