prove that every integer is either even or odd but never both.

Please examine and make an assessment on Ford over the last five years compared to the auto industry? [200 words or more][Will give thumbs up]

a) y''(x)-3y'(x)=8e^3x+4sinx

b) y''(x)+y'(x)+y(x)=0

c) y^(iv)(x)+2y''(x)+y(x)=0

Matrix:

Ax b
[2 1 0 0 0 | 100]
[1 1 -1 0 -1 | 0]
[-1 0 1 0 1 | 50]
[0 -1 0 1 1 | 120]
[0 1 1 -1 1 | 0]

Problem 5
Compute the solution to the original system of equations by transforming y into x, i.e., compute x = inv(U)y.
Solution:
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I have not Idea how to do this. Please HELP!

The Goodsmell perfume company has a new line of perfume and is designing a new bottle for it. Because of the expense of the glass required to make the bottle, the surface area must be less than 150 cm2. The company also wants the bottle to contain at least 100mL of perfume. The design under consideration is in the shape of a cylinder. Determine the maximum volume possible for a cylindrical bottle that has a total surface area of less than 150 cm2. Determine the volume to the nearest 10mL. Report the dimensions of the bottle and the corresponding surface are and volume.

let p = 1031, Find the number of solutions to the equation x^2 -2 y^2=1 (mod p), i.e., the number of elements (x,y), x,y=0,1,...,p-1, which satisfy x^2 - 2 y^2=1 (mod p)

Consider the following graphs. Select the graphs that are planar. There are 3 correct answers out of 5.

A. The wheel graph, W6.

B. The complete bipartite graph, K3,3.

C. The 4-cube graph, Q4.

D. The complete graph, K4.

E. The cycle graph, C5.

Use power series approximations method to approximate the solution of the initial value problem: y"− (1+ x) y = 0 y(0) = 1 y'(0) = 2 (Write all the terms up to the power ). x^4

(a). Prove the law of quadratic reciprocity for odd primes. Include all definitions and results used throughout the proof.

(b). Explain why it is enough to prove the reciprocity law for 2 and the quadratic reciprocity law for odd primes to answer the general question: Is m a square mod n for any positive integers m, n?

(c). Describe the role of permutations in the proof of quadratic reciprocity.

Solve each item supporting your answer with clear explanation.

1.Consider the following augmented matrix of a system of linear equations.

( 7 −3 4 6

−3 2 6 2

2 5 3 −5 )

a. Solve the system with the Jacobi method. First rearrange to make it diagonally dominant if possible. Use [0,0,0] as the starting vector. Find the condition number of the matrix of coefficients κ_∞(A), and compute how many iterations are required to get the solution accurate to five significant digits?

b. Repeat part a) using the Gauss-Seidel method. Are fewer iterations required?

c. Is convergence faster in parts a) and b) if the starting vector is [-0.26602, -0.26602,-0.26602]?

Find the approximations L_n, R_n, T_n, and M_n for n = 5, 10, and 20. Then compute the corresponding errors E_L, E_R, E_T, and E_M. (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.)

If an undamped spring-mass system with a mass that weighs 24 lb and a spring constant 16 lbin is suddenly set in motion at t=0 by an external force of 432cos(8t) lb, determine the position of the mass at any time. Assume that g=32 fts2. Solve for u in feet.

Enclose arguments of functions in parentheses. For example, sin(2x).

u(t) = ?

3. Solve using probabilistic dynamic programming: I would like to sell my computer to the highest bidder. I have studied the market, and concluded that I am likely to receive three types of offers: an offer of $200 with probability 2/7, and offer of $300 with probability 4/7, and an offer of $400 with probability 1/7. I will advertise my computer for up to three consecutive days. At the end of each of the three days, I will decide whether or not to accept the best offer made that day. What is the optimum strategy for maximizing the expected sale price for my computer? What is this expected sale price?