Use the Laplace transform to solve the following initial value problem: y′′−3y′−28y=δ(t−7) y(0)=0, y′(0)=0

y(t)=

Let G and H be groups. Define the direct product

G X H = {(x,y) | x ∈ G and y ∈H }

Prove that G X H itself is a group.

find the stedy-periodic solution cap of the initial value problem
x"+2x'+50x=6cos(4t),x(0)=0,x'(0)=0

Suppose $90,000 was invested on January 1, 1980 at an annual effective interest rate of 4% in order to provide an annual (calendar-year) scholarship of $5,000 each year forever, the scholarships paid out each January 1.

(a)

In what year can the first $5,000 scholarship be made?

(b)

What smaller scholarship can be awarded the year prior to the first $5,000 scholarship? (Round your answer to the nearest cent.)

b belongs to N. Prove that if {7m, m belong to Z} is not belongs to {ab, a is integer}, then b =1

Let T: R^2 -> R^2 be an orthogonal transformation and let A is an element of R^(2x2) be the standard matrix of T. In a) and b) below, by rotation we mean "rotation of R^2 by some angle and the origin". By reflection, we mean "reflection of R^2 over some line through the origin".

a) Show that T is either a rotation or a reflection.

b) Show that every rotation is a composition of 2 reflections, and thus that T is a composition of at most 2 reflections.

Solve each initial value problem, then classify the type and stability of the critical point at (0,0) of each system:

x1'=4x1-5x2; x2'=5x1-4x2

x1(2019)=16; x2(2019)=25

Solve each initial value problem, then classify the type and stability of the critical point at (0,0) of each system:

x1'=-3x1+5x2; x2'=-2x1-x2

x1(0)=-7; x2(0)=3

A car rental agency in a major city has a total of 2900 cars that it rents from three locations: Metropolis Airport, downtown, and the smaller City Airport. Some weekly rental and return patterns are shown in the table (note that Airport means Metropolis Airport).
Rented from
Returned to   AP   DT   CA
Airport (AP)   90%   10%   10%
Downtown (DT)   5%   80%   5%

At the beginning of a week, how many cars should be at each location so that the same number of cars will be there at the end of the week (and hence at the start of the next week)?
AP     cars
DT     cars
CA     cars

Devise a procedure between a seller of a house and a buyer of the house that will decide if the sellers minimal acceptable price is below or equal to the buyers maximal acceptable price without revealing the acceptable prices to the other side of the negotiations.

find a particular solution to
y"-2y'+y=-12.5e^t

In a sample of 61 business trips taken by employees in the HR department, a company finds that the average amount spent for the trips was $1480 with a standard deviation of $450. In a sample of 101 trips taken by the employees in the sales department is $1650 with a standard deviation of $550. When testing the hypothesis (at the 5% level of significance) that the average amount spent on trips taken by the sales department are higher than those taken by the HR department what is the null and alternative hypotheses? (please write out the null and alternative hypotheses below AND on your scratch work...failure to do BOTH will result in 0 points)

Consider the Sturm-Liouville problem
X′′(x) + λX(x) = 0 subject toX′(0) = 0, X(l) = 0.

1. Are the boundary conditions symmetric?

2. Do these boundary conditions yield negative eigenvalues?
3. Determine the eigenvalues and eigenfunctions, Xn(x). (It is enough in some cases to provide the equation that determines the eigenvalues rather than an explicit formula.)
4. Are the eigenfunctions orthogonal?

Design a Root Finding solver in Matlab that satisfies the arguments below;
• Inputs will be coeficients of the equation.
• Outputs will be roots
• It should work with any polynomial
?=?0+?1?1+?2?2+?3?3+⋯+????