Solve the following initial value problem ut + 2ux − 2u = e t+2x , u(x, 0) = 0

Prove that for every n ∈ N:

a) (10^n + 3 * 4^(n+2)) ≡ 4 mod 19, [note that 4^3 ≡ 1 mod 9]

b) 24 | (2*7^(n) + 3*5^(n) - 5),

c) 14 | (3^(4n+2) + 5^(2n+1) [Note that 3^(4n+2) + 5^(2n+1) = 9^(2n)*9 + 5^(2n)*5 ≡ (-5)^(2n) * 9 + 5^(2n) *5 ≡ 0 mod 14]

A) Using the scenario described by “Each store employs one or more employees; each employee is employed by one store”, work the following problems: 1. Identify each entity. 2. Identify each relationship type. Justify your answer. 3. Create the basic Crow's Foot ERD using Visio.

B) Using the scenario described by “A job assignment can be assigned to more than one employee at a time; each employee can have only one job assignment”, work the following problems: 1. Identify each entity. 2. Identify each relationship type. Justify your answer. 3. Create the basic Crow's Foot ERD using Visio

Prove that R(4; 4) = 18.

I am required to make at least three (3) substantive posts in this discussion. Your initial response of at least 150 words. I need to describe two (2) uses for quadratic equations. Provide your own original examples.

11. Solve numerically the following Boundary Value Problem.

X²Y” – X(X+2)Y’ + (X+2)Y = 0

Y(1) = e and Y(2) = 2e²

The value of e = 2.71828

Let⇀F and⇀G be vector fields defined on R3 whose component functions have continuous partial derivatives. Furthermore, assume that ⇀∇×⇀F=⇀∇×⇀G.Show that there is a scalar function f such that ⇀G=⇀F+⇀∇f.

Calculate each of the following (or indicate why it is not defined) (a) (1,2,3,4) x (4,3,2,1) (b) (1,1,1) x [(1,2,3) x (3,3,0)] (C) (1,1,1). [(1,2,3) x (3,3,0) (d) (1,1,1) x [(1,2,3). (3,3,0)] (e) (1,1,1) x [(1,2,3) - (3,3,0)] (f) (1,1,1) + [(1,2,3) X (3,3,0)]

This is a Combinatorics Problem

Consider the problem of finding the number of ways to distribute 7 identical pieces of candy to 3 children so that no
child gets more than 4 pieces. Except Stanley (one of the 3 children) has had too much candy already, so he’s only
allowed up to 2 pieces. Write a generating function & use your generating function to solve this problem.

You invest your $2500 COVID-19 stimulus check into 3 separate accounts. The Certificate of Deposit (CD) pays 4% interest annually. The stock market pays 5% annually. The toilet paper hoarding business pays 10% annually. If the amount spent on the toilet paper hoarding business is $400 more than you spent on the CD, and the total interest gained is $169, how much money did you invest in each venture. Please show all your equations, and the matrix you used to solve this.

List some examples explaining how quadratic equations are used in business.

For each of the following polynomials, determine the number of the roots that are in the LHP, in the RHP, and on the jw-axis.

i. s³ + s + 2 = 0

ii. s⁴ - 4s³ + 7s² - 8s + 10 = 0

** When solving for LHP, RHP, & jw-axis, can you explain what differentiates these different categories? (i.e. I understand that the RHP you need to look for a sign switch but do not understand the LHP and jw-axis) Thank you!

Solve x'' + x' -6x = 5e^tsint using variation of parameters.

Q1: Let x = [ x(t) y(t) ] and consider the system of ODEs

x' = [5/2, 3; −3/4 ,−1/2] x.

(1)

1.1 Solve the initial value problem subject to x(0) = 1, y(0) = 1.