Solve the initial value problem below using the method of Laplace transforms.

ty''-4ty'+4y=20, y(0)=5 y'(0)=-6

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. (Please show clear steps and explain them)

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with nitrous oxide. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 pounds/sq.in., the nitrous oxide chamber inside the rocket will explode. Tiff worked from a formula p=14.7e−h/10p=14.7e−h/10 pounds/sq.in. for the atmospheric pressure hh miles above sea level. Assume that the rocket is launched at an angle of αα above level ground at sea level with an initial speed of 1400 feet/sec. Also, assume the height (in feet) of the rocket at time tt seconds is given by the equation y(t)=−16t2+1400sin(α)ty(t)=−16t2+1400sin⁡(α)t. [UW]

a. At what altitude will the rocket explode?
b. If the angle of launch is αα = 12∘∘, determine the minimum atmospheric pressure exerted on the rocket during its flight. Will the rocket explode in midair?
c. If the angle of launch is αα = 82∘∘, determine the minimum atmospheric pressure exerted on the rocket during its flight. Will the rocket explode in midair?
d. Find the largest launch angle αα so that the rocket will not explode.

please show some shortcut tricks and technique formula for integration to solve out easy difficulties questions
I would give a postive rating..if you help me a little

Finding Surface Area In Exercises 43-46, find the area of the surface given by z = f(x, y) that lies above the region R.
f(x,y)=4-x^2 R: triangle with vertices (-2,2),(0,0),(2,2)

For what values of x this set is linearly independent?

{sin(x), cos(x), x}.

Check the true statements below:

A. The orthogonal projection of y onto v is the same as the orthogonal projection of y onto cv whenever c≠0.

B. If the columns of an m×n matrix A are orthonormal, then the linear mapping x→Ax preserves lengths.

C. If a set S={u1,...,up} has the property that ui⋅uj=0 whenever i≠j, then S is an orthonormal set.

D. Not every orthogonal set in Rn is a linearly independent set.

E. An orthogonal matrix is invertible.

Solve the following problem:

y ''' − 2y '' + y ' = 2 − 24e^ x + 40e ^5x

y(0) = 1/2 , y' (0) = 5/2 , y'''(0) = − 9/2

BOTH LINES ARE PART OF A SYSTEM OF EQUATIONS

Please show the calculation process in Excel, thank you

7. With the gasoline time series data from Table 8.1, show the exponential smoothing forecasts using α=0.1.

   1. Applying the MSE measure of forecast accuracy, would you prefer a smoothing

      constant of α=0.1 or α=0.2 for the gasoline sales time series?

   2. Are the results the same if you apply MAE as the measure of accuracy?

   3. What are the results if MAPE is used?

   4. Week	Sales (1000s of gallons)
      1	17
      2	21
      3	19
      4	23
      5	18
      6	16
      7	20
      8	18
      9	22
      10	20
      11	15
      12	22

Solve the given differential equation by undetermined coefficients. y'' − 6y' + 9y = 21x + 3

Let V = R4 and let U = hu1, u2i, where u1 =    1 2 0 −3    , u2 =     1 −1 1 0    . 1. Determine dimU and dimV/U. 2. Let v1 =    1 0 0 −3    , v2 =     1 2 0 0    , v3 =     1 3 −1 −6    , v4 =     −2 2 0 9    . For any two of the vectors v1,...,v4, determine whether they are in the same coset of U in V or not. 3. Find a basis of V that contains a basis of U. Hence, determine a basis of V/U. 4. Find two (distinct) elements of the coset e1 + U.

a) What is the difference between regression and interpolation?
b) Use least squares regression to fit a straight line to the data given in Table 1 and
calculate the y value corresponding x=3.
c) Find the Lagrange interpolating polynomial using the data given in Table 1 and
calculate the y value corresponding x=3.

Table 1

This question pertains to the probability of rolling a given total (adding the two die face-values) using 2 fair 6-sided dice. The possible results of rolling 2 fair 6-sided dice are the elements of the Cartesian product A × A, where A = {1, 2, 3, 4, 5, 6}.

(a) How many elements does A × A have?

(b) Partition the elements (x, y) of the set A × A according to the sum x + y. For example, (1, 3) goes in the part labeled somehow by ‘4.’

(c) Write down all of the parts of the partition with all of their respective elements.

(d) Define an equivalence relation R on A × A so that the equivalence classes of R equal the parts of the partition in Part (c).

(e) Write down one representative of each equivalence class in the equivalence relation R in Part (d).

(f) The most likely die total is |P|/|A × A|, where P is the largest part of the partition you wrote down in Part (c). What is the most likely die total?

Use the Laplace transform to solve the given initial-value problem.

y'' - 2y'' - 8y = 2sin2t; y(0) = 2, y'(0) = 4

Find the coordinates of the images of A (2,3) and B (-2,3) under the following transformations. Assume that all dilations are centered at the origin. Draw sketches to justify your answer.

a. A dilation with a scale factor of 3 followed by a dilation with a scale factor 2.

b. A dilation with scale factor 2 followed by a translation with a slide arrow from (2,1) to (3,4).

c. A translation with a slide arrow from (2,1) to (3,4) followed by a dilation with scale factor 2.