solve the given DE or IVP (Initial-Value Problem).

a. 2y′ + y cot x = 8y^-1 cos³ x

b. y′ = sin² (3x − 3y + 1)

c. xy′ + y ln x = y ln y

d. x² dy/dx = y² + 5xy + 4x²

Your task is to build a wastewater treatment plant to handle all of the domestic wastewater produced by a housing estate. Using population equivalent, PE = 65000, design, describe and calculate briefly the sizes of all the unit processes in the treatment plant that are going to be built. You may assume the value that is not given.

Kirchhoff’s voltage law states that the sum of the voltage drops across a resistor, R, an inductor, L, and a capacitor, C, in an electrical circuit must be the same as the voltage source, E(t), applied to that RLC circuit. Applying the additional fact that the current, I, is related to the charge, q, by the relationship I=dq/dt, the resulting ODE model for the charge in a circuit is: L (d^2 q)/(dt^2 )+R dq/dt+1/C q=E(t). If a 100 sin⁡(60t)source is connected to an RLC circuit with a 1/20 henry inductor, L, a 1 ohm resistor, R , and a 1/130 farad capacitor, C, find the charge q(t) given that q(0)=0 and I(0)=q'(0)=0.

What is the number of comparisons in the bubble sort algorithm, if it is used to sort a list of n-entries? Justify your formula.

Use the simplex method to solve the linear programming problem.

A company manufactures Products A, B, and C. Each product is processed in three departments: I, II, and III. The total available labor-hours per week for Departments I, II, and III are 900, 1080, and 840, respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows. (For example, to make 1 unit of product A requires 2 hours of work from Dept. I, 3 hours of work from Dept. II, and 2 hours of work from Dept. III.)

How many units of each product should the company produce in order to maximize its profit?

What is the largest profit the company can realize?
$

Are there any resources left over? (If so, enter the amount remaining. If not, enter 0.)

Using the pigeonhole theorem prove that any set of 220 10-character strings over the alphabet {a,b,c,d} contains a pair of anagrams.

(1 point) Use the method of undetermined coefficients to find a solution of

y′′−16y′+144y=48e8tcos(9t)+48e8tsin(9t)+5e0t

Use a and b for the constants of integration associated with the homogeneous solution. Use a as the constant in front of the cosine term.
y=yh+yp=

(1)Prove that for every a, b ∈ R, |a + b| = |a| + |b| ⇐⇒ ab ≥ 0. Hint: Write |a + b| 2 = (|a| + |b|) 2 and expand.

(2) Prove that for every x, y, z ∈ R, |x − z| = |x − y| + |y − z| ⇐⇒ (x ≤ y ≤ z or z ≤ y ≤ x). Hint: Use part (1) to prove part (2).

True or False

Every proper subspace of R² can be visualized as being either {(0,0)} or a line through the origin in R².

(PDE)

Write the soln using separation of variables , in the form of fourir series:

Utt=Uxx

boundary: U(t,0)=0=U(t,pi)

initial :

initial: U(0,x)=1 and Ut(0,x)=0

Your Christmas ski vacation was great, but it unfortunately ran a bit over budget. All is not lost, because you just received an offer in the mail to transfer your $12,600 balance from your current credit card, which charges an annual rate of 20.4 percent, to a new credit card charging a rate of 11 percent.

(a) How much faster could you pay the loan off by making your planned monthly payments of $255 with the new card? (Do not round intermediate calculations and round your final answer to 2 decimal places (e.g., 32.16).)

Number of months? _____

(b) What if there was a 2 percent fee charged on any balances transferred? (Do not round intermediate calculations and round your final answer to 2 decimal places (e.g., 32.16).)

Number of months? _____

Solve using Laplace Transform:

1) y'' - 2y' + 5y = cos(2t) - cos(2t)u_4pi(t); y(0) = 0, y'(0) = 0

Universal Mines Inc. operates three mines in West Virginia. The ore from each mine is separated into two grades before it is shipped. The daily production capacities of the three mines, as well as their daily operating costs, are as follows:

Universal has committed itself to deliver 54 tons of high-grade order and 65 tons of low-grade ore by the end of the week. Universal can run its mines seven days a week if required.

Determine the number of days each mine should be operated during the upcoming week if Universal Mines is to fulfill its commitment at the minimum total cost. Round the answers to two decimal places.

a. The number of days Mine I should operate =                            [ Select ]                       ["2.65", "5.94", "1.75", "1.00"]         days

b. The number of days Mine II should operate =                            [ Select ]                       ["3.95", "7.00", "4.06", "5.23"]         days

c. The number of days Mine III should operate =                            [ Select ]                       ["6.00", "7.00", "2.95", "5.00"]         days

d. The total cost of the operation for next week = $                            [ Select ]                       ["279,000", "284,000", "320,000", "245,000"]      

How many equations would you need to compute a series of cubic splines of the form ax^3 + bx^2 + cx + d to fit through 4 data points and why?