Explain the process for modeling a linear spring in a static system using F = k*s

What is an example of a vector space besides Rⁿ?

What are the three properties of a vector subspace?

Question 16:

What is the general solution of the following homogeneous second-order differential equation?

Non-integers are expressed to one decimal place.

d^2y/dx^2 − 11.y = 9

Question 17:

What is the general solution of the following homogeneous second order differential equation?

d^2y/dx^2 + 3dy/dx − 4.y = cos(3.x)

Question 18:

What is the general solution of the following homogeneous second order differential equation?

d^2y/dx^2 + 13dy/dx + 40.y = sin(2.x)

Question 19:

What is the general solution of the following homogeneous second order differential equation?

d^2y/dx^2 + 1dy/dx − 20.y = 1.x − 3

Question 20:

What is the general solution of the following homogeneous second order differential equation?

d^2y/dx^2 − 10dy/dx + 29.y = 1.e(3.x)

Construct a generator matrix and a parity check matrix for a ternary Hamming code Ham(2, 3).

Assume a codeword x from for the ternary Hamming code Ham(2, 3)$ was sent and the word y was received. Use the partiy check matrix you constructed in part (a) to decode y in each part
using syndrome decoding:

(b) y = ( 1 , 1 , 1 , 0 ),

(c) y = ( 2 , 2 , 2 , 2 ),

(d) y = ( 1 , 2 , 1 , 2 ).

A 6 lb weight is placed upon the lower end of a coil spring suspended from a fixed beam. The weight comes to rest in its equilibrium position, thereby stretching the spring 4 inches. Then
beginning at t = 0 an external force given by F (t) = 27 sin(4t) − 3 cos(4t) is applied to the system. The resistance of the medium is three times the velocity (in ft/sec). Find the displacement of the weight as a function of time.

Suppose an object falls from a great height on a planet where the constant of the acceleration of gravity is g = 7.84. Assume that the resistance of the atmosphere is proportional to the square of the velocity of the object with constant of proportionality k = 0.25. Establish and solve an Initial Value Problem to express the velocity of the object as a function of time. Find the terminal velocity of the object. Graph this function. Then express the fall distance as a function of time. Graph this function. Hint: g − kv2 = (√g − (√k)*v) (√g + (√k)*v)

prove that a group of order 9 is abelian

Water is boiled in a bowl and cooled in a room. The air temperature in the room is increasing linearly according to the function Ta(t) = 30 + 0.01t (t in minutes, T in ∘C. Assume that Newton's Law of Cooling is satisfied: the rate of change of the temperature of the water is proportional to the difference between the temperature of the water and the temperature of the environment. We take the temperature of the water after 10 minutes and find that it is 81∘C. Explain and graph Ta. Establish and solve an Initial Value Problem to express the water temperature as a function of time, graph this function.

1. a. Consider d on R, the real line, to be d(x,y) = |x² – y²|. Show that d is NOT a metric on R.    b.Consider d on R, the real line, to be d(x,y) = |x³ – y³|. Show that d is a metric on R.

   2. Let d on R be d(x,y) = |x-y|. The “usual” distance. Show the interval (-2,7) is an open set.

Note: you must show that any point z in the interval has a ball centered at z, and that ball is completely contained within the interval (-2,7).

(a) Show that x= 0 is a regular singular point.

(b) Find the indicial equation and the indicial roots of it.

(c) Use the Frobenius method to and two series solutions of each equation

x^2y''+xy'+(x^2-(4/9))y=0

(1 point) A tank is filled with 1000 liters of pure water. Brine containing 0.05 kg of salt per liter enters the tank at 6 liters per minute. Another brine solution containing 0.07 kg of salt per liter enters the tank at 9 liters per minute. The contents of the tank are kept thoroughly mixed and the drains from the tank at 15 liters per minute. A. Determine the differential equation which describes this system. Let S(t) denote the number of kg of salt in the tank after t minutes.

A) dS/dt = ?

B) Solve the differential equation for S(t).

A marketing research firm provides you with the following information. Historically, they have correctly predicted a positive market 82% of the time and correctly predicted a negative market 76% of the time. Without any market survey information, the estimate for a favorable market is 50% and an unfavorable market is 50%.

a)  What is the probability (in percentage) of a favorable market, given that the market survey predicts a favorable market? Answer in integer value.

b)  What is the probability (in percentage) of an unfavorable market, given that the market survey predicts an unfavorable market?  Answer in integer value.

Determine if the following sets are real vector spaces with the indicated operations.

(a) The set V of all ordered pairs (x, y) of real numbers with the addition defined by (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2 + 1) and scalar multiplication defined by α(x, y) = (αx, αy + α − 1), α ∈ R

(b) The set V of all 2 × 2 real matrices X with the addition of M22 but scalar multiplication ∗ defined by α ∗ X = αXT , α ∈ R

Solve the following system :

z” + y ′ = cos x,

y” − z = sin x,

z(0) = −1, z′ (0) = −1, y(0) = 1, y′ (0) = 0.