Show that the indicial roots of the singularity x=0 differ by an integer. Use the method of Frobenius to obtain the first four terms of at least one series solutions of the DE:

xy''+2y'-xy=0

A tank originally contains 120 gal of fresh water. Then water containing 1/2 lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 11 min the process is stopped, and fresh water is poured into the tank at a rate of 2 gal/min, with the mixture again leaving at the same rate. Find the amount of salt Q(11) in the tank at the end of an additional 11 min.

Round your answer and intermediate answer to two decimal places.

Amount of salt in the tank at the end of an additional 11 min is Q(11)= _____ lbs.

Expand the function, f(x) = x, defined over the interval 0 <x <2, in terms of:
A Fourier sine series, using an odd extension of f(x)

and A Fourier cosine series, using an even extension of f(x)

4) Laplace Transform and Solving second order Linear Differential Equations with Applications The Laplace transform of a function, transform of a derivative, transform of the second derivative, transform of an integral, table of Laplace transform for simple functions, the inverse Laplace transform, solving first order linear differential equations by the Laplace transform

Applications: a) Series RLC circuit with dc source b) Damped motion of an object in a fluid [mechanical, electromechanical] c) Forced Oscillations [mechanical, electromechanical]

You should build the theory portion of your report on what you have learned in the math courses including Mathematics 1, Applied Mathematics, Differential calculus and Integral Calculus. Any additional material you need in order to begin or complete your project must be included and discussed within the report. Special attention is paid to the consistency of the derivation of formulae and concepts in your report. Once the mathematical foundations are laid in a proper way, you need to introduce the topic you have been assigned from your program where the mathematical concepts you have explored are applied. It is important you support your derivations and conclusions by real world engineering applications

Dean takes his boat out fishing every weekend. His current boat is still in okay condition, but he decides he’d like to buy a new one. He finds the boat of his dreams for $20,950. He does some research and finds that his credit union will give him a 5-year loan with an APR of 4.25% if he makes a down payment of 18%.

1. What is the amount of Dean’s down-payment?   

2. What is the loan amount that Dean will be financing?

3. What will his monthly payments be on the loan?

4. What is the total amount Dean will pay for his boat if he finances the purchase?

5. How much interest will Dean pay the credit union over the life of the loan?   

Sam, Dean’s younger brother, tries to convince Dean to save up for a new boat instead of getting a loan. He tells Dean that the credit union has a savings account option that offers an APR of 1.2% compounded monthly, so long as the account maintains a minimum balance of $2000.

6. Since Dean has the down payment amount already, if he uses that to start a savings account with the credit union, then adds $200 to the account each month, how much will he have at the end of 5 years?

7. If Dean starts the savings account with his down payment money, how much would he have to deposit each month for 5 years to end up with the $20,950 he needs for the boat of his dreams?

8. If Dean follows the savings plan from problem 7, how much total will he have deposited into the savings account at the end of the 5 years?

9. If Dean follows the savings plan from problem 7, how much interest will he have earned in the savings account over the course of the 5 years?

10. Compare the total Dean would pay on the loan to the total Dean would deposit into the savings account. How much would Dean save by going with the savings option?

11. If Dean decides he wants to save up for the boat as fast as possible, how long would it take the account to reach the full value of $20,950 if he uses the down payment money to start the account then adds $400 each month? Give the answer as years with 2 decimal places!

Suppose V and V0 are finitely-generated vector spaces and T : V → V0 is a linear transformation with ker(T) = {~ 0}. Is it possible that dim(V ) > dim(V0)? If so, provide a specific example showing this can occur. Otherwise, provide a general proof showing that we must have dim(V ) ≤ dim(V0).

A Professor decides to hide their hoard money, 100 identical gold coins, in 20 unique locations hidden across the city. It is assumed that multiple coins or even none can be stored at each location.

How many ways can he distribute these coins if neither the Mining Lab location nor the Office location can have more than 20 coins (e.g., 25 coins to the Mining Lab and 25 coins to the Office with 50 coins to other locations would not be legal)? It is assumed they cannot trust any of his colleagues with their precious and hard-earned research funds.

a) How many integers in between 1 and 10⁶ have an even number of divisors?

Show work proving your answer.

Express your answer in prime factorized form.

b) With proof, determine all integer solutions to the following equation:

1935x + 2322y = 177

a symmetric group S5 acts on the set X5 = {(i, j) : i, j ∈ {1, 2, 3, 4, 5}}.

S5 will also act on this set. Consider the subgroup H = <(1, 2)(3, 4), (1, 3)(2, 4)>≤ S5. (a) Find the orbits of H in this action. Justify your answers. (b)

For each orbit find the stabiliser one of its members. Justify your answers.

action is this t.(i,j)=(t(i),t(j))

Mass Springs Systems problem (Differential Equations)

A mass weighing 6 pounds, attached to the end of a spring, stretches it 6 inches.

If the weight is released from rest at a point 4 inches below the equilibrium position, the system is immersed in a liquid that imparts a damping force numerically equal to 3 times the instantaneous velocity, solve:

a. Deduce the differential equation which models the mass-spring system.
b. Calculate the displacements of the mass ? (?) at all times “?”
c. Make a graph that shows the motion

Thanks for the help

Trevor, imprisoned in a cell, wishes to get a message to his friend, Franklin who is loitering outside a wall surrounding his prison. Trevor has wrapped his message round a stone, and intends to throw it from his prison cell window, over the wall, to land where Franklin can retrieve it. Trevor can throw the stone at 16 m/s. The cell window from which he can throw the stone is 6 m above the level ground outside the prison. The wall rises 10 m above the ground outside the prison, and is at a horizontal distance of 20 m from Trevor’s cell window. The stone has mass 0.2 kg.

a) Suppose that Trevor throws the stone at an angle of 2π / 9 radians (40 degrees) above the horizontal. Describe whether the stone will clear the wall, neglecting air resistance.

b) Analyze what range of launch angles will enable Trevor to get the stone over the wall, if air resistance can be neglected?

c) Trevor wants to throw the stone so that it lands as far as possible from the wall, to reduce the chances of Franklin being detected as he searches for it. Describe what launch angle should he choose (still neglecting air resistance and stone must clear the wall)? For this launch angle, analyze how far beyond the wall will the stone land?

Demand for rug-cleaning machines at Clyde’s U-Rent-It is shown in the following table. Machines are rented by the day only. Profit on the rug cleaners is $19 per day. Clyde has 3 rug-cleaning machines.

  
a. Assuming that Clyde’s stocking decision is optimal, what is the implied range of excess cost per machine? (Enter smaller value in first box and larger value in second box. Do not round intermediate calculations. Round your answers to 2 decimal places. Omit the "$" sign in your response.)

Implied range of excess cost per machine from $  to $

b. Your answer from part a has been presented to Clyde, who protests that the amount is too low. Does this suggest an increase or a decrease in the number of rug machines he stocks?

• Increase

• Decrease

c. Suppose now that the $19 mentioned as profit is instead the excess cost per day for each machine and that the shortage cost is unknown. Assuming that the optimal number of machines is 3, what is the implied range of shortage cost per machine? (Enter smaller value in first box and larger value in second box. Do not round intermediate calculations. Round your answers to 2 decimal places. Omit the "$" sign in your response.)

Implied range of shortage cost per machine from $  to $