While using laplace transforms, solve the following diff eq

x'' + 6x' + 25x = 0

with initial conditions:

x(0) = 2 and x'(0) = 3

The temperature function (in degrees Fahrenheit) in a three dimensional space is given by T(x, y, z) = 3x + 6y - 6z + 1. A bee is constrained to live on a sphere of radius 3 centered at the origin. In other words, the bee cannot fly off of this sphere. What is the coldest temperature that the bee can experience on this sphere? Where does this occur? What is the hottest temperature that the bee can experience on this sphere? Where does this occur?

The processing division of the Sunrise Breakfast Company must produce one ton (2000 pounds) of breakfast flakes per day to meet the demand for its Sugar Sweets cereal. Cost per pound of the three ingredients is: Ingredient A $4 per pound Ingredient B $3 per pound Ingredient C $2 per pound Government regulations require that the mix contain at least 10% ingredient A and 20% ingredient B. Use of more than 800 pounds per ton of ingredient C produces an unacceptable taste. Determine the minimum-cost mixture that satisfies the daily demand for Sugar Sweets.

During the course of a Friday night, a nightclub often receives some counterfeit ten-dollar bills. At one point in the night, there are two counterfeit ten-dollar bills randomly distributed in a stack of a total of 8 ten-dollar bills (counterfeit and legitimate) in the cash register. From that point on, no additional ten-dollar bills are received, they are only paid out from the top as a change to patrons of the nightclub. What is the probability that the nightclub will have no counterfeit ten-dollar bills in its cash register if only 4 ten-dollar bills are paid out during the night?

1. What is the definition of an eigenvalue and eigenvector of a matrix?

2. Consider the nonhomogeneous equationy′′(t) +y′(t)−6y(t) = 6e2t.

(a)Find the general solution yh(t)of the corresponding homogeneous problem.

(b)Find any particular solution yp(t)of the nonhomogeneous problem using the method of undetermined Coefficients.

c)Find any particular solution yp(t)of the nonhomogeneous problem using the method of variation of Parameters.

(d) What is the general solution of the differential equation?

3. Consider the nonhomogeneous equationy′′(t) + 9y(t) =9cos(3t).

(a)Findt he general solution yh(t)of the corresponding homogeneous problem.(b)Find any particular solution yp(t)of the nonhomogeneous problem.(c) What is the general solution of the differential equation?

4. Determine whether the following statements are TRUE or FALSE.Note: you must write the entire word TRUE or FALSE. You do not need to show your work for this problem.(

a)yp(t) =Acos(t)+Bsin(t)is a suitable guess for the particular solution ofy′′+y= cos(t).(

b)yp(t) =Atetis a suitable guess for the particular solution ofy′′−y=et.

(c)yp(t) =Ae−t2is a suitable guess for the particular solution ofy′′+y=e−t2.(d) The phase portrait of any solution ofy′′+y′+y= 0is a stable spiral.

5. Consider the matrixA=[−2 0 0,0 0 0,0 0−2].(

a) Find theeigenvaluesofA.

(b) Find theeigenvectorsofA.

(c) Does the set of all the eigenvectorsofAform a basis ofR3?

6. Consider the system of differential equationsx′(t) =−2x+y,y′(t) =−5x+ 4y.

a) Write the system in the form~x′=A~x.

b) Find the eigenvalues of.

c) Find theeigenvectorsofA.

d) Find the general solution of this system.

e) Sketch the phase portrait of the system. Label your graphs.

7. Determine whether the following statements are TRUE or FALSE. You must write the entire word “TRUE” or “FALSE’’. You do not need to show your work for this problem.

a) If|A|6= 0 then A does not have a zero eigenvalue.

(b) IfA=[4 2,0 4]then the solution ofx′=Axhas a generalized eigenvector of A.

(c) LetA=[−1 4 0,0 3 3,1 0−2].The sum of the eigenvalues of A is 18.

(d) Let x′=Ax be a 2x2 system. If one of the eigenvalues of A is negative, the stability structure of the equilibrium solution of this system cannot be a stable spiral.

8. Below (next page) are four matrices corresponding to the 2x2 system of equations x′=Ax,where x= (x1, x2). Match each of the four systems (1)–(4) with its corresponding vector field, one of the four plots (A)–(D), on the next page. You do not need to show your work for this problem.

A=[0 1,1−1]

A=[0−1,1 0]

A=[1 2,−2 1]

A=[−1 0,−1−1]

Find y as a function of x if y^(4)−6y′′′+9y′′=0,

y(0)=7, y′(0)=11, y′′(0)=9, y′′′(0)=0.

Show that any graph with n vertices and δ(G) ≥ n/2 + 1 has a triangle.

1. Let G be a bipartite graph with 107 left vertices and 20 right vertices. Two vertices u, v are called twins if the set of neighbors of u equals the set of neighbors of v (triplets, quadruplets etc are defined similarly).

   Show that G has twins.
   Bonus: Show that G has triplets. What about quadruplets, etc.?

Using the Chinese remainder theorem solve for x:

x = 1 mod 3

x = 5 mod 7

x = 5 mod 20

Please show the details, I`m trying to understand how to solve this problem since similar questions will be on my exam.

Robotix manufacture three robots: Mavis, Charles and Koala; each with different capabilities. All three require special circuits, of which up to 1,000 can be obtained each week. Mavis takes three of them, Charles four of them and Koala six of them. Work is limited to 400 hours per week. The construction of each Mavis consumes two working hours, Charles one hour and Koala three hours. Profits are $500, $250 and $400 respectively for each Mavis, Charles and Koala that is sold. The Robotix has signed a contract with a major customer to make and supply at least 100 Mavis, 100 Charles and 30 Koala each week.

(a) Set up the liner programming problem (variables, objective function, constraints, etc.) Use Excel Solver to answer the following Question:

(b) How many of each robot should Robotix make per week to maximize total profit? What is the maximum profit?

(c) If the profit per Mavis is increased to $510, does the optimal solution change and why? What should be the new optimal profit?

(d) With overtime, the company may increase the working hours to 460 hours. Should the company do that? How much more they can get in profit?

(e) Should the company increase the quantity of each robot? Explain.

11.1      Determine the matrix inverse for the following system:

10x₁ + 2x₂ − x₃ = −27

−3x₁ −6x₂ +2x₃ = −61.5

x₁   + x₂   +5x₃ = −21.5

Check your results by verifying that [A][A]^-1 = [I ]. Do not use a pivoting strategy.

Calculate A^-1 using the LU decomposition of A. Use the Matlab lu command to perform the LU decomposition. Use the Matlab backslash \ operator to perform intermediate linear system solutions. Use Matlab matrix multiplication to verify that A^-1 was correctly calculated.

(§3.4 # 3) Use the previous problem to show that the average number of binary
comparisons required to sort n items is at least O(n log2 n).

would you tell me some list of topic that is not mathematics but can be discussed how mathematics can be used to understand?

something like egnima or turing machine, the golden ratio

Additional Problem on the mapping of w=exp z: Find the image of the semi-infinite strip " x\le 0, and -\pi/2 \le y < pi/2" under the map w=exp z, and label corresponding portions of the boundaries. Here, "\le" means less than or equal to.