Please answer all parts of the following question. Please show all work and all steps.

1a.) Solve the initial value problem W((t^2) + 1, f(t)) = 1, f(0) = 1

1b.) Let x1 and x2 be two solutions of x''+ ((x')/(t)) + q(t)x = 0, t > 0, where q(t) is a continuous function. Given that W(6)=7, find W(7)

1c.) Show that any solution of x''+ 5x' + 6x = 0 tends to zero as t approaches positive infinity.

1d.) Solve x'' + 2x' = 0, x(0) = 0, the limit as t approaches positive infinity x(t) = a

1e.) Solve x'' - x = t(e^(2t))

Honeydew bottles honey jars and sells them through retail channels. The weight on the sticker says 20 oz and Honeydew claims its bottles have a weight that is normally distributed with a mean of 20 oz and std dev of 2 oz. The retailer has been receiving several complaints lately and decides to measure a sample of 4 jars and finds weights of 18, 20, 17 and 19 oz respectively.
a. The retailer complains to Honeydew if he is 95% confident that the mean is lower than advertised. Does he complain?
b. If the retailer’s customers return any jar under 18 oz in weight, what fraction of the retailer’s sales result in a return?

c. Assuming Honeydew’s claim is true, what is their current sigma capability? What can they do to reduce the fraction of returns to 5%? What then would the new sigma capability be?

I WANT AN ABSOLUTELY CORRECT ANSWER WITH DETAILS.
PLS VERIFY BEFORE YOU POST

1. Let (X,d) be a metric space.

1. Show that every open d-ball is a d-open subset of X
2. Show that every closed d-ball is a d-closed subset of X.

2: Let (X,d) be a metric space. Show that a subset A of X is d-open if and only if it is the union of a (possibly empty) set of open d-balls.

Prove using mathematical induction: 3.If n is a counting number then 6 divides n^3 - n. 4.The sum of any three consecutive perfect cubes is divisible by 9. 5.The sum of the first n perfect squares is: n(n +1)(2n +1)/ 6

Suppose P, Q and R are atomic propositions.

(a) Show that the conjunction connective satisfies the commutative and associativity property.

(b) Show that the disjunction connective satisfies the commutative and associativity property.

(c) Construct a propositional form using all three atomic propositions above as well as the connectives conjunction, disjunction and conditional.

(d) Construct an equivalent propositional form for (c).

y''' −2y' −4y = 0, y(0) = 6, y'(0) = 3, y''(0) = 22

solve the initial value problem

You would convert it to m^3-2m-4=0. You find the root (m=2) and use synthetic division to find the other roots. m^2+2m+2 is what you get. I am stuck on what to do next?

y = 2e^(−x)*cosx−3e^(−x)*sinx + 4e^(2x) is the answer.

FlorU football programs are printed 1 week prior to each home game. Attendance averages 90,000 screaming and loyal Tators fans, of whom two-thirds usually buy the program, following a normal distribution with standard deviation of 5000 programs. A program sells for $4 each. Unsold programs are sent to a recycling center that pays 10 cents per program. The cost to print each program is $1.

a. How many programs should be ordered per game to maximize expected profit?

b. What is the stockout risk for this order size?

c. How sensitive is the order quantity in (a) to the following estimates?

I. The standard deviation of demand

II. The selling price of a program

III. The cost of recycling an unsold program Answer by considering at least 4 values of each quantity and calculating the corresponding order quantity. Draw a graph showing the effect of the above variables with order quantity on the Y axis. Comment briefly (1-2 lines) on each graph.

Sketch the graph of the following function: f(x) = x²+5x/25-x²
Make sure each solution has the following information with STEP BY STEP
Domain of f(x).
x-intercepts and y -intercepts. If x-intercepts are hard to compute, then ignore
them.
Vertical asymptotes.
Horizontal asymptotes.
Intervals where f is increasing and decreasing.
Local minima and local maxima.
Intervals where f is concave up and concave down.
Inflection points

You can use two types of fertilizer in your orange grove, Best Food and Natural Nutri. Each bag of Best Food contains 8 pounds of nitrogen, 4 pounds of phosphoric acid, and 2 pounds of chlorine. Each bag of Natural Nutri contains 3 pounds of nitrogen, 4 pounds of phosphoric acid and 1 pound of chlorine. You know that the grove needs at least 1,000 pounds of phosphoric acid and at most 400 pounds of chlorine. If you want to minimize the amount of nitrogen added to the grove, how many bags of each type of fertilizer should be used? How much nitrogen will be added?

Find f(1), f(2), f(3) and f(4) if f(n) is defined recursively by f(0)=4f(0)=4 and for n=0,1,2,…n=0,1,2,… by:
(a) f(n+1)=−2f(n)
f(1)=
f(2)=
f(3)=
f(4)=

(b) f(n+1)=4f(n)+5
f(1)=
f(2)=
f(3)=
f(4)=

(b) f(n+1)=f(n)²−4f(n)−2
f(1)=
f(2)=
f(3)=
f(4)=

Write a function or script that will solve linear systems of any size by Gaussian elimination with partial pivoting in Python.

6. Find all monic irreducible polynomials of degree ≤ 3 over Z3. Using your list, write each of the following polynomials as a product of irreducible polynomials over Z3:

(a) x^4 + 2x^2 + 2x + 2.

(b) 2x^3 − 2x + 1.

(c) x^4 + 1.

GRAPH THEORY

Prove/Show that a connected Graph G is not separable if and only if it is nonseparable.

Definitions for Reference: A connected Graph G is called nonseparable if it has no cut vertices (A vertex v in a connected graph G is caled a cut vertex if G-v is disconnected)

A connected graph G is called separable if there exist subgraphs H1, H2 ⊂ G. with E(H1) ∪ E(H2) = E(G) and E(H1) ∩ E(H2) = ∅. V (H1) ∪ V (H2) =V (G) and V (H1) ∩ V (H2) containing a single vertex.

A plane in R3 can be given by an equation Ax + By + Cz = D where A, B, C, and D areconstants and A, B, and C are not all zero. Suppose two planes are given by equations A1x+B1y+C1z = D1 and A2x+B2y+C2z = D2. The intersection of the two planes can be empty, or it can be a line, or the planes could be identical. How can the correct possibility be determined from the constants in the equations? Explain! (Hint: Think in terms of solving linear systems.)

Math of Finance

Alex is paying 1184.87 in monthly mortgage payments. She wants to refinance her existing mortgage loan of $100,000 at 14% interest for 30 years that she obtained 4 years ago. Her mortgage officer informed her that he could get her a rate of 10% but the finance cost would include paying a prepayment penalty on the existing loan, which is equal to 6 months interest on the balance of the loan, plus a closing cost of $2000. Would it be worth if for Alex to refinance?