find the general solution of the given differential equation

1. 2y''+3y'+y=t^2 +3sint

find the solution of the given initial value problem

1. y''−2y'−3y=3te^2t, y(0) =1, y'(0) =0

2.  y''−2y'+y=te^t +4, y(0) =1, y'(0) =1

Let S1 be the level surface given by the equation f(x,y,z) =k1and letS2be the level surface given by the equation g(x,y,z) =k2. Suppose that C is the curve of intersection of the surfacesS1andS2andassume that P is a point on C. show that any tangent vector to C at P must be parallel to∇f(P)×∇g(P)

Solve each of the following linear systems by the method of the example on page 283.

(a) 3x − y = 4        x + 5y = 9

(b) x − 2y + z = 1       y + z = 0         −x + y + z =−3

HInt; first find the QR decomposition of the coefficient matrix via the Gram–Schmidt process

Use ten iterations of the appropriate MATLAB function, with x^(0)=[0,...,0]', to solve Ax=b (approximately).

A)use Jacobi iteration.

B) use Gauss-siedel iteration.

1) make sure to use SOR with w=1.25, w=1.5, w=1.75,w=1.9, and optimal value if given.

* A=[1,-2,0,0;-2,5,-1,0;0,-1,2,-0.5;0,0,-0.5,1.25]] , B=[-3;5;2;3.5]. , (optimal w is 1.5431.)

A) Use Jacobi or Gauss-Seidel iteration and perform three iterations by hand.

B) Use Jacobi or Gauss-Siedel iteration for ten iterations with a MAT-LAB function.

* A= [5, -1,0;-1,5,-1;0,-1,5] , B=[9;4;-6]

Allow the main process to generate a text file containing the text of this assignment.

The main process will then create two other processes and pass to them the name of the file and two codes (code1 and code2) using a pipe() system call. The codes are two different alphabetic letters such as “a” and “k”.

• The child processes should search the file for the character in the interval received, compute their frequencies and return them through a separate pipe to the parent process.

• The parent process should compute the total number of characters in the file, and the rate of the appearance of the character frequencies received from the two child process, through separate pipes.

• The parent process should then form a table of alphabetic characters and their frequencies and print the table in a proper format on the screen.

Hint: You are expected to use fork(), pipe(), read(), write(), close(), waitpid(), and other appropriate system call, if needed.

Use ten iterations of the appropriate MATLAB function, with x^(0)=[0,...,0]', to solve Ax=b (approximately).

B) use Gauss-siedel iteration.

C)use SOR with w=1.25, w=1.5, w=1.75,w=1.9, and optimal value if given.

* A=[4,8,0,0;8,18,2,0;0,2,5,1.5;0,0,1.5,1.75] , B=[8;18;0.5;-1.75]. , (optimal w is 1.634.)

6. Suppose K1 and K2 are compact. Why is K1 ∪ K2 necessarily also compact?

(a) Write a proof of this using the sequential definition.

(b) Write a proof of this using the “closed and bounded” definition.

(c) Write a proof of this using open covers and subcovers.

Problem 2: You have applied to two universities X and Y for graduate admission. There is a .25 chance that you are going to get into X and .20 chance that you are going to get into Y. Keep in mind that admission into the universities is independent events.

a)    What is the probability that you are going to get into both universities?

b)     What is the probability that you are going to get into at least one university?

c)    What is the probability that you are going to get into any one of the university?

d)    Are getting into those universities mutually exclusive?

In a simple undirected graph H the sum of vertex degrees is 60. What is the smallest possible number of vertices in this graph? What is the largest possible number of vertices in the graph?

• What do you know about how math and historical developments are related? Share any information you already have about the topic.
• Are there any questions you still have?
• Do you believe that the teaching and learning of math has changed?
• What are some of your first conjectures of how math affected the growth of civilization?

Assume we are flipping 1000 fair coins. What can you say about the probability that you will get at least 3/5 of them to be heads? Compare the 3 bounds (Markov, Chebyshev, Chernoff), and try plugging in some leargevalue of n to see how muchs they differ!

You need a loan of $160,000 to buy a home. Calculate your monthly payments and total closing costs for each choice below.

Choice 1: 30 year fixed rate at 5% with closing costs if $2300 and no points.

Choice 2: 30 year fixed rate at 4.5% with closing costs of $2300 and 5 points.

What is the monthly payment for choice one?

What is the monthly payment for choice two?

what is the total closing cost for choice one?

what is the total closing cost for choice two?

why might choice one be better?

why might choice two be better?

The next three questions are based on the following information.

To study the weight accuracy of a 50lb fertilizer bag, 12 samples of 12 bags of fertilizer in each sample were taken and the results are as follows.

1.

The overall average weight of a bag of fertilizer is ________ pound, and the average range is ____________ pound. (Please round to two decimal points and include no units.)

2.

The upper control limit for a 99.7% control chart for the mean is ________ pound, and the lower control limit is __________ pound. (Please round to two decimal points and include no units. Please enter the upper limit first.)

3.  

The upper control limit for a 99.7% control chart for the range is_________ pound, and the lower control limit is _________ pound. (Please round to two decimal points and include no units. Please enter the upper limit first.)

Explain Egyptian arithmetic.