using the Laplace transform solve the IVP

y'' +4y= 3sin(t) y(0) =1 , y'(0) = - 1 , i am stuck on the partial fraction decomposition step. please explain the decomposition clearly.

1. Find the distance between the skew lines with parametric equations x = 1 + t, y = 3 + 6t, z = 2t, and

                 x = 1 + 2s, y = 6 + 15s, z = −2 + 6s.

2. Find the equation of the line that passes through the points on the two lines where the shortest distance is measured.

Given the following 40x40 matrix below, with starting vector V shown below also, apply the power method to find the dominant eigenvalue of matrix using MATLAB program, MAPLE program or some other computer program to print out: the estimate of the lambda with tolerance 0.01, the number of iterations, and the converged lambda. Then print out the transpose of the eigenvector (should be 40 components) produced with up to two decimals.

Starting Vector V= [

Matrix A =

[

4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

-1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 0 0 0 0 0 -1 4 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

-1 0 0 0 0 0 0 0 0 0 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0;

0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0;

0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 0 0 0 0 0 0 0 0 0 -1;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 4 -1 0 0 0 0 0 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4 -1;

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 4;

]

Write a short essay approximately 2 pages (for a first year engineering class).Introduce the concept of eigenvalues and eigenvectors including a short history of their development. Outline in detail some of the real life applications of the use of eigenvalues and eigenvectors in science, engineering and computer science, giving at least one example in each case. 10 Marks

Provide an example

1) A nested sequence of closed, nonempty sets whose intersection is empty.
2) A set A that is not compact and an open set B such that A ∪ B is compact.

3) A set A that is not open, but removing one point from A produces an open set.

4) A set with infinitely many boundary points.

5) A closed set with exactly one boundary point

Is there a difference between the means of occupancy rates in May and August? Answer your question by calculating an approximate and appropriate symmetric 95% confidence interval using a Z statistic. Explain your findings

The Chinese Remainder Theorem for Rings.

Let R be a ring and I and J be ideals in R such that I + J = R. (a) Show that for any r and s in R, the system of equations x ≡ r (mod I) x ≡ s (mod J) has a solution. (b) In addition, prove that any two solutions of the system are congruent modulo I ∩J. (c) Let I and J be ideals in a ring R such that I + J = R. Show that there exists a ring isomorphism R/(I ∩J) ∼ = R/I ×R/J.

The length of the essay does not exceed 800 words in APA format

Discuss the fundamentals of factorial design including main effects and interaction effect using a 2 × 2 factor design. Use your own hypothetical example for illustration.

Solve the following LP using revised simplex algorithm in table format.

Min 6x1+10x2 -18x3 +25x4 +15x5

subject to 0.2x1+ 0.2x2+0.4x3+ 0.5x4 + x5 ≥ 1000

2.5x1+ 1.5x2+ x3 + 0.5x4 + 0.5x5 ≤ 1100

0.8x3 + x5 ≥ 500

x1, x2, x3, x4, x5 ≥ 0

Prove that the solution of the discrete least squares problem is given by the orthogonal projection.

Consider the following relations:

R₁ = {(a, b) ∈ R² ∣ a > b}, the greater than relation
R₂ = {(a, b) ∈ R² ∣ a ≥ b}, the greater than or equal to relation
R₃ = {(a, b) ∈ R² ∣ a < b}, the less than relation
R₄ = {(a, b) ∈ R² ∣ a ≤ b}, the less than or equal to relation
R₅ = {(a, b) ∈ R² ∣ a = b}, the equal to relation
R₆ = {(a, b) ∈ R² ∣ a ≠ b}, the unequal to relation

For these relations on the set of real numbers, find

R2∘R1

R2∘R2

R3∘R5

R4∘R1

R5∘R3

R3∘R6

R4∘R6

R6∘R6

Use your graphing calculator to find the solutions to the following equation for

0° ≤ θ < 360°

by defining the left side and right side of the equation as functions and then finding the intersection points of their graphs. Make sure your calculator is set to degree mode. (Round your answers to one decimal place. Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)

3 sin² θ + 1 = 5 sin θ

Use Laplace transforms to solve the initial value problem:

y''' −y' + t = 0,

y(0) = 0,

y'(0) = 0,

y''(0) = 0.

Did any of your previous assignments involve math calculations for premium and or salary analysis and adjustments? Please provide detailed example(s).