Y''+y'-20y=xe^3x+e^4x

Find the general solution of this differential equation

why the cylindrical shape of a can is the optimal shape

Needs to be 2 pages.

Solve the following linear integer programming model using the Cutting Plane method. Show all relevant work in your solution report.

Maximize Z = x1 + x2

Subject to

3x1 + 2x2 < 5

x2 < 2

x1, x2 > 0 and integer.

Show that every permutational product of a finite amalgam am(A,B: H) is finite.Hence show that every finite amalgam of two groups is embeddable in a finite group.

A country's census lists the population of the country as 254 million in 1990, 286 million in 2000, and 314 million in 2010. Fit a second-degree polynomial passing through these three points. (Let the year 2000 be x = 0 and let p(x) represent the population in millions.)

p(x) = ________million

Use this polynomial to predict the populations in 2020 and in 2030.

2020_________million

2030 ________million

Show the following identities for a, b, c ∈ N.

(a) gcd(ca, cb) = c gcd(a, b) Hint: To show that two integers x, y ∈ Z are equal you can show that both x | y and y | x which implies x = y or x = −y. Thus, if both x and y have the same sign, they must be equal.

(b) lcm(ca, cb) = c lcm(a, b)

(c) ab = lcm(a, b) gcd(a, b) Hint: Consider first the case that gcd(a, b) = 1 and show that ab = lcm(a, b) in this case. For the general case combine this with (b).

(d) lcm(gcd(a, c), gcd(b, c)) = gcd(lcm(a, b), c) Hint: First treat the special case that gcd(a, b, c) = 1. In this case begin by showing that lcm(gcd(a, c), gcd(b, c)) = gcd(a, c) gcd(b, c). The asserted equality gcd(a, c) gcd(b, c) = gcd(lcm(a, b), c) is then shown by proving that gcd(a, c) gcd(b, c)| gcd(lcm(a, b), c) and gcd(lcm(a, b), c)| gcd(a, c) gcd(b, c). Proceed to show that gcd(a, c)| lcm(a, b) and gcd(a, c)| c, and deduce from this that gcd(a, c)| gcd(lcm(a, b), c); proceed analogously for gcd(b, c). Then argue that gcd(a, c) gcd(b, c)| gcd(lcm(a, b), c) under the present assumption. Conversely, in order to show that gcd(lcm(a, b), c)| gcd(a, c) gcd(b, c), write according to (a) gcd(a, c) gcd(b, c) = gcd(gcd(a, c)b, gcd(a, c)c) = gcd(gcd(ab, bc), gcd(ac, c2 )), and show that gcd(lcm(a, b), c) divides all of ab, bc, ac, and c 2 . Explain from here why gcd(lcm(a, b), c) must divide gcd(a, c) gcd(b, c) then as well. For the general case explain how (a) and (b) can be used to reduce the general assertion to the previously treated special case.

***The only help I really need is with c and d. I just added a and b for context.

Here are four questions:

1. Prove a standard Brownian motion is Gaussian process .

2. Prove a Brownian bridge is Gaussian process.

3. Prove Ornstein-Uhlenbeck process is Gaussian

4. Prove the position process is Gaussian.

Please provide as many detail as you can, thanks.

(Topology) Prove that the interior of a subset A of Xτ is the union of all τ -open sets contained in A.

Explanation:
Assume the reader understands derivatives, and knows the definition of instantaneous velocity (dx/dt), and knows how to calculate integrals but is struggling to understand them. Use students’ prior knowledge to provide an explanation that includes the concept and physical meaning of the integral of velocity with respect to time.

Reminder: The user is comfortable with the calculations, but is struggling with the concept. To fully address the prompt, emphasize the written explanation in English over the calculation

Prove that if a sequence is bounded, then it must have a convergent subsequence.

Given a set S = {1, -1, i, -i} where i² = -1 and with multiplication * on this set,

1. complete the following Cayley table:

(b) Determine whether, or not, the operation * is a binary operation on S.

(c) Is * commutative on S?

(d) Investigate the following properties of binary operations for this operation on S:

i. Closed

ii. Identity

iii. Inverse

iv. Associativity

Respond to the following prompts in a minimum of 175 words:

• Consider a recent interaction you have had with a co-worker or item that you saw in the media that might have included logical errors.
• Share the example as well as the logical errors that are present in the example. Be sure to use terms and concepts from the content covered this week to support your discussion.

Alice and Bob are have several piles of chips. On each turn they can either remove 1 or 2 chips from one pile, or split a pile into two nonempty piles. Players take turns and a player that cannot make a move loses. Find the value of the Sprague–Grundy function for positions with one pile made of n chips. (Please do not forget to prove correctness of your asnwer.)