Use the Laplace transform to find the solution of the IVP:

a.) 2y' + y = 1, y(0) = 2 (answer should be y(t) = 1 + e^{-t / 2} )

f.) 4y" + y = 0, y(0) = -1, y'(0) = -1 (answer should be y(t) = -sin(t) - cos(t))

Please show work!

What is the difference between multiplying a matrix times a vector and multiplying two matrices?

For Exercises 1-4 below, (a) verify that y1 and y2 satisfy the given second-order equation, and (b) find the solution satisfying the given initial conditions (I.C.).

2. y′′−3y′+2y=0; y1(x)=e^x,y2(x)=e^2x. I.C.y(0)=0,y′(0)=−1.
3. y′′−2y′+y=0; y1(x)=e^x,y2(x)=xe^x. I.C.y(0)=1,y′(0)=3.

Let a sequence {xn} from n=1 to infinity satisfy

x_(n+2)=sqrt(x_(n+1) *xn) for n=1,2 ......

1. Prove that a<=xn<=b for all n>=1

2. Show |x_(n+1) - xn| <= sqrt(b)/(sqrt(a)+sqrt(b)) * |xn - x_(n-1)| for n=2,3,.....

3. Prove {xn} is a cauchy sequence and hence is convergent

Please show full working for 1,2 and 3.

The post office uses a multiple channel queue (M/M/C), where customers wait in a single line in front of two service providers for the first available window. If the average service time is 1 minute and the arrival rate is 7 customers every five minutes, find, the average number of people waiting in line?

The standard error of the estimate for the above bivariate data is:

Question 3 options:

Show that if 100 people of different heights stand in a line, it is possible that we find neither 11 people of increasing heights nor 11 people of decreasing heights (describe a counterexample) (Hint: Apply the Erdos-szekeres theorum)

3. Suppose that in any given period an unemployed person will find a job with probability .6 and will therefore remain unemployed with a probability of .4. Additionally, persons who find themselves employed in any given period may lose their job with a probability of .3 and will have a .7 probability of remaining employed.

   1. Set up the Markov transition matrix for this problem

   2. There are 1000 people in the economy. At period 0, half of the

      population is unemployed, what will be the number of unemployed

      people after 1 period?

   3. What will be the number of unemployed people after 4 period?

   4. What is the steady-state level of unemployment?

find the number of ways to distribute 18 balls, three each of six different colors, into three boxes

1. Consider the set of real numbers S=[-1,0)U{1/n:n=1,2,3,...}. Give the explicit set for each of the

1. The complement of S,
2. The closure of S,
3. The boundary points of S,
4. The limit points of S,
5. The isolated points of S,
6. The interior of S, and
7. The exterior of S.

Explain in simple terms what a primitive root is for prime modulus, and show that 2 is a primitive root of 11, but 3 is not.

Find the series solution of the following equation x(d^2y/dx^2+(1-x)dy/dx+ny=0. where n=0,1,2

According to Zeller's Formula, the day of the week can be calculated by:

f=k+(13m−1)/5+D+D/4+C/4−2C

where:

•k is the day of the month.

•m is the month number designated in a special way: March is 1, April is 2, . . . , December is10; January is 11, and February is 12. If x is the usual month number, i.e. for January x is 1, for February x is 2, and so on; then m can be computed with this formula:m= (x+21)%12+1,where % is the usual modulus (i.e. remainder) function. Alternatively,m can be computed in this way:

m= x+10, if x≤2, or x−2 otherwise.

•D is the last two digits of the year, but if it is January or February those of the previous year are used.

•Cis for century, and it is the first two digits of year. In our example,C=20.

•From the result f we can obtain the day of the week based on this code:

Day0 = Sunday...Day 6 = Saturday

For example, if=123, thenf%7=4, and thus the day was Thursday. Again, % is the modulus function.

What was the day on June, 1 2005 according to Zeller's formula?

1. There are two career fairs held at a Catholic Chicago University.

1. Fill in the not-hired counts for each fair.

2. Do the samples meet the Success-Failure Conditions?

3. What are the proportions of hired students for each fair?

4. What are the individual standard errors for hired students?

5. What is the difference of proportions of hired students between the two fairs?

6. What is the joint standard error of the two fairs?

7. What is the z-score associated with 90 confidence interval?

8. Compute the confidence interval for this difference of proportions.

1. Which fair should a student attend?

Please solve all parts of the following question. Please show all work and all steps.

1a.) Solve

x' = x + 3y + 2t

y' = x - y + t^2

1b.) Solve

x' + ty = -1

y' + x' = 2

1c.) Solve

x' + y = 3t

y' - tx' = 0