Given the differential equation

y''+y'+2y=0,  y(0)=−1,  y'(0)=2y′′+y′+2y=0,  y(0)=-1,  y′(0)=2

Apply the Laplace Transform and solve for Y(s)=L{y}Y(s)=L{y}. You do not need to actually find the solution to the differential equation.

6) Consider the differential equation y^JJ + by^J + 16y = 0 For which value(s) of b does the solution

I. decay rapidly to 0 as t → ∞

II. oscillate regardless of t value

III. decay while oscillating

a. For b = 10 and y(0) = 0, y^J(0) = 6, solve the initial value problem.

Consider the initial value problem: y0 = 3 + x−y, y(0) = 1 (a) Solve it analytically. (b) Solve it using Euler’s method using step size h = 0.1 and find an approximation to true solution at x = 0.3. (c) What is the error in the Euler’s method at x = 0.3

Manny was arrested as the main suspect in a murder investigation. Manny was brought into an interrogation room, but was not read his Miranda rights. A police officer asked Manny, “Where were you on the night of May 2?” Manny proceeded to confess to the murder, detailing the time and location of the murder and the use of a knife as a murder weapon.

The officer immediately left the room, and he then realized that he accidentally forgot to read Manny his Miranda rights. He reported his mistake to the chief, who decided to place Manny in a holding cell.

Two hours later, Manny was brought back into the interrogation room where he was questioned by a different officer. Prior to questioning, this officer properly read Manny his Miranda rights.

Manny stated that he understood his rights and repeated his confession about the time and location of the murder but did not mention the knife.

Police then went to the location and found the victim and the weapon (a knife) used in the murder next to the body. The knife was later found to have Manny's blood and fingerprints.

At his trial, Manny admitted that both of his statements were voluntarily made, but argued that they should each be suppressed, along with the murder weapon (knife) since all were the "fruit" of a Miranda violation.

Which, if any, pieces of evidence should be suppressed?

Creatine and protein are common supplements in most bodybuilding products.  Bodyworks, a nutrition health store, makes a powder supplement that combines creatine and protein from two ingredients (X1 and X2). Ingredient X1 provides 20 grams of protein and 5 grams of creatine per pound.  Ingredient X2 provides 15 grams of protein and 3 grams of creatine per pound.  Ingredients X1 and X2 cost Bodyworks $5 and $7 per pound, respectively.  Bodyworks wants its supplement to contain at least 30 grams of protein and 10 grams of creatine per pound and be produced at the least cost.

Determine what combination will maximize profits.

Use Excel Solve to determine the solution:

Amazon regularly couriers rectangular packages overseas. The girth of a rectangular package is defined to be the perimeter of a cross section perpendicular to the length.You prefer to use Speedy Couriers for your international deliveries, but they will only carry rectangular packages where the sum of length and girth is at most 150 cm. Find the dimensions of the package with the largest volume that they will carry. Assume that the critical point gives a maximum

Solve the differential equation by variation of parameters, subject to the initial conditions

y(0) = 1, y'(0) = 0.

2y'' + y' − y = x + 7

Consider the initial value problem X′=AX, X(0)=[−4,-2], with A=[−6,0,1,−6] and X=[x(t)y(t)] (a) Find the eigenvalue λ, an eigenvector V1, and a generalized eigenvector V2 for the coefficient matrix of this linear system. λ= , V1= ⎡⎣⎢⎢ ⎤⎦⎥⎥ , V2= ⎡⎣⎢⎢ ⎤⎦⎥⎥ $ (b) Find the most general real-valued solution to the linear system of differential equations. Use t as the independent variable in your answers. X(t)=c1 ⎡⎣⎢⎢ ⎤⎦⎥⎥ + c2 ⎡⎣⎢⎢ ⎤⎦⎥⎥ (c) Solve the original initial value problem. x(t)= y(t)=

If f(x)=sign(x−2)+|x+2| for (-4,4] and you extend f(x) to a periodic function on the real line, and F(x) is the Fourier series of f(x). Which of the following options are correct? (Select all that apply.)

I. F(1)=2.

II. F(0)=1.

III. f(x) is continuous on the interval.

IV. F(x) is an odd function.

V. F(x) is continuous on the interval.

Pirates are attacking the coastal town of Townberg. The pirates have

a cannon on their ship which can fire a cannonball at 440 meters per

second.

(a) Suppose the pirates fix the cannon at an angle of π/4. Write a function h(t) which gives the height (in meters) at t seconds for the cannonball (assume that the cannonball has initial height 5 meters) and a function d(t) which gives the horizontal distance of the cannonball (in meters) at t seconds.

(b) How far from Townberg does the pirate ship have to be so that

the cannonball can hit the town (assume Townberg has a height

of 0 meters)?

(c) What is the cannonball’s vertical speed as it hits the ground?

What is the cannonball’s horizontal speed as it hits the ground?

What is the cannonballs total speed as it hits the ground?

ICE13B - “How long will the money last?” You begin on 1 January 2015 by making an initial deposit of $5,000 to open a savings account at Bank of Mason. This account pays 0.135% interest per month. Every even numbered month, you withdraw $500 to pay your tuition installment balance. At the end of December each year you receive a $2,000 end-of-year bonus from your employer which you deposit into your account. Run your simulation until your account balance goes negative. Question 1: During what month does you account balance go negative after making the withdrawal? Question 2: Prepare a plot showing your account for all months A Matlab code is needed

(Advanced Calculus and Real Analysis) - Lebesgue outer measure

* Please prove that the Cantor set C has Lebesgue outer measure zero.