An urn always contains two balls, where each ball is either red or blue.

At each stage a ball is randomly chosen from the urn. A drawn red
ball is always replaced with a blue ball. A drawn blue ball is equally
likely to be replaced by either a red or a blue ball. Suppose that the
urn initially has one red and one blue ball.
(a) Define a Markov chain that should be useful for the above model.
Define its states and give the transition probabilities.
(b) Find the probability that the second ball selected is red.
(c) Find the probability that the third ball selected is red.
(d) Find the long run proportion of time that both balls are red.
(e) Find the long run proportion of drawn balls that are red.

initially a large tank with a capacity of 300 gallons contains 150 gallons of clean water. A solution of salt with a concentration of 2 lb / gal flows into the tank at a rate of 50 gal / min. The solution is perfectly well mixed while the solution is extracted at a rate of 25 gal / min. Find:

a) the amount of salt in the tank at the time it is filled (in pounds)

b) the speed at which salt is coming out at this time (in pounds per minute)

c) The amount of salt that has left the tank since the beginning and until that moment (in pounds)

The given table shows the preference schedule for an election with 5 candidates. Find the complete ranking using the method of pairwise comparisons.​ (Assuming that ties are broken using the results of the pairwise comparisons between the tying​ candidates.)

Choose the answer below

( ) E,D,B,C,A

( ) A,B,C,D,E

( ) E,A,B,C,D

( ) D,C,E,A,B

We say that a set of system specifications is consistent if there is a way for all the specifications to be simultaneously true. Otherwise, the set of specifications is inconsistent. Consider the following set of system specifications:

If the file system is not locked, then new messages will be queued.

The file system is not locked if and only if the system is functioning normally.

If new messages are not queued, then they will be sent to the message buffer.

If the file system is not locked, then new messages will be sent to the message buffer.

New messages will not be sent to the message buffer.

Use propositional logic to determine if the set of system specifications consistent or inconsistent by doing the following:

Part a (2 pts) : First, define the appropriate propositional variables for representing the system specifications.

Part b (2.5 pts) : Then, translate each of the system specifications into propositional logic using the propositional variables you defined in part a.

Part c (2.5 pts) : Finally, is the set of system specifications consistent or inconsistent? Justify your answer.

4) Write a brief reflection of Task 2 and 3 which may include description on data type used to solve the given task, variable used, and objects created. Reflection should also include justification on logic used to solve the given task along with proper references

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

y'' − 7y' = 12e^6t − 6e^−t,    y(0) = 1, y'(0) = −1

Find the following Taylor Series with given centers. Use the algebraic methods rather than the derivatives.

1. f(x) = x^3 + x^2 -2x +3 centered at a= -1

2. sin(x) centered at a= pi/2

3. f(x) = (e^x - e^-x)/2 centered at a= 0

A firm has prepared the following binary integer program to evaluate a number of potential locations for new warehouses. The firm’s goal is to maximize the net present value of their decision while not spending more than their currently available capital.

Max 20x1 + 30 x2 + 10x3 + 15x4

s.t. 5x1 + 7x2 + 12x3 + 11x4 ≤ 21 {Constraint 1}

x1 + x2 + x3 + x4 ≥ 2 {Constraint 2}

x1 + x2 ≤ 1 {Constraint 3}

x1 + x3 ≥ 1 {Constraint 4}

x2 = x4 {Constraint 5}

x j ={ 1, if location j is selected 0, otherwise xj=1, if location j is selected0, otherwise

Solve this problem to optimality and answer the following questions:

A. What is the net present value of the optimal solution? (Round your answer to the nearest whole number.)

B. How much of the available capital will be spent (Hint: Constraint 1 enforces the available capital limit)? (Round your answer to the nearest whole number.)

a. Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation that the coefficients must satisfy.

b. Find the first four nonzero terms in each of two solutions y1 and y2 (unless the series terminates sooner).

y''-xy'-y=0 ; x0=0

A 1-kg mass stretches a spring 20 cm. The system is attached to a dashpot that imparts a damping force equal to 14 times the instantaneous velocity of the mass. Find the equation of motion if the mass is released from equilibrium with an upward velocity of 3 m/sec.

SOLVE THIS USING MATLAB CODE (SECOND ORDER DIFFERENTIAL EQUATIONS)

SOLVE THIS USING MATLAB CODE (SECOND ORDER DIFFERENTIAL EQUATIONS)

SOLVE THIS USING MATLAB CODE (SECOND ORDER DIFFERENTIAL EQUATIONS)

SOLVE THIS USING MATLAB CODE (SECOND ORDER DIFFERENTIAL EQUATIONS)

SOLVE THIS USING MATLAB CODE (SECOND ORDER DIFFERENTIAL EQUATIONS)

SOLVE THIS USING MATLAB CODE (SECOND ORDER DIFFERENTIAL EQUATIONS)

SOLVE THIS USING MATLAB CODE (SECOND ORDER DIFFERENTIAL EQUATIONS)

SOLVE THIS USING MATLAB CODE (SECOND ORDER DIFFERENTIAL EQUATIONS)

Fibonacci numbers are defined by F0 = 0, F1 = 1 and Fn+2 = Fn+1 + Fn for all n ∈ N ∪ {0}.

(1) Make and prove an (if and only if) conjecture about which Fibonacci numbers are multiples of 3.
(2) Make a conjecture about which Fibonacci numbers are multiples of 2020. (You do not need to prove your

conjecture.) How many base cases would a proof by induction of your conjecture require?

y� � y � 2x sin x

y'+xy=x³+y²    Solve the differential equation.

Answer the following questions.
(a) What is the implication of a correlation matric not being positive-semidefinite?
(b) Why are the diagonal elements of a correlation matrix always 1?
(c) Making small changes to a positive-semidefinite matrix with 100 variables will have no effect on the matrix. Explain this statement?

Solve this Initial Value Problem using the Laplace transform:

x''(t) - x'(t) - 6x(t) = e^(4t),

x(0) = 1, x'(0) = 1