Find the general solution of the ODE:

y'' − 6y' + 9y = (1 + x^2)e^2x .

In: Advanced Math

a. Let A be a square matrix with integer entries. Prove that if lambda is a rational eigenvalue of A then in fact lambda is an integer.

b. Prove that the characteristic polynomial of the
companion matrix of a monic polynomial f(t) equals f(t).

In: Advanced Math

I have the following discrete data.

x=[missing, missing, missing, 758, 763, 742, 729, 721, 714, 709, 696, 680 ]

y=[87.5, 86.4, 84.5, 83.6, 83.2, 84.0 83.2, 82.6, missing, 82.0, 81.2, 80.8 ]

Use cubic spline to predict the y-value at a x-value of 735 (using carefully commented matlab code).

Please present the theory behind the mathematics of the model. I prefer if you do all this as handwritten notes

In which range could the validity of the model be and why?

In: Advanced Math

Two candles have the same length. One candle takes 6 hours to burn all the way down while the other candle takes 9 hours. If the two candles are lit at the same time, how long will it take for the two candles to burn so that one candle is twice the length of the other?

In: Advanced Math

An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will be normally distributed with μ=4 and σ2=4, whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with parameters (6, 9). A point is randomly chosen on the image and has a reading between 4 and 5. What is the probability that the point is in the black region? Give your answer in terms of Φ.

Assume that α = 1/4. The answer is .1720 for your reference; please make sure you are getting the correct answer. Please show your work and explain your answer.

In: Advanced Math

There are a total of 10 test stations. When a circuit board fails, the test station will automatically record the lifetime for the failed board and then immediately switch to a replacement board. The test was stopped at 1200 hours. At that time, 19 circuit boards had failed and there was an operating circuit board in every station. Assuming the boards have exponentially-distributed lifetimes, estimate the mean lifetime for this type of board.

In: Advanced Math

How do the concepts of master production scheduling and material requirements planning translate to service organization? Provide an example.

In: Advanced Math

Let G = Z3 × Z6 × Z2.

(a) What is the order of (2, 3, 1) in G?

(b) Find all the possible orders of elements of G. Is the group G cyclic?

In: Advanced Math

The phase difference between sine and cosine function is 90 degrees

true or false

Phase shift of a sinusoid function can be either positive or negative.

true or false

In: Advanced Math

1)

A) Write **v** as a linear combination of
**u**_{1}, **u**_{2}, and
**u**_{3}, if possible. (Enter your answer in
terms of **u**_{1},
**u**_{2}, and **u**_{3}.
If not possible, enter IMPOSSIBLE.)

**v** = (−1, 7, 2), **u**_{1}
= (2, 1, 5), **u**_{2} = (2, −3, 1),
**u**_{3} = (−2, 3, −1)

B) Write **v** as a linear combination of
**u** and **w**, if possible, where
**u** = (1, 3) and **w** = (2, −1).
(Enter your answer in terms of **u** and
**w**. If not possible, enter IMPOSSIBLE.)

**v** = (−3, −9)

C) Find **w** such that 2**u** +
**v** − 3**w** = **0**.

**u** = (0, 0, −6, 2),

**v** = (0, −3, 5, 1)

In: Advanced Math

3. Solve the following linear programming problem. You must use the dual. First write down the dual maximization LP problem, solve that, then state the solution to the original minimization problem.

(a) Minimize w = 4y_{1} + 5y_{2} +
7y_{3}

Subject to: y_{1} + y_{2} + y_{3} ≥
18

2y_{1} + y_{2} + 2y_{3} ≥ 20

y_{1} + 2y_{2} + 3y_{3} ≥ 25

y_{1}, y_{2}, y_{3} ≥ 0

(b) Making use of shadow costs, if the 2^{nd} original
constraint changed to

2y_{1} + y_{2} + 2y_{3} ≥ 24, now what
will the minimum of w be? Explain clearly.

(c) Making use of shadow costs, if the 1^{st} original
constraint changed to

y_{1} + y_{2} + y_{3} 21, now what will
the minimum of w be? Explain clearly.

In: Advanced Math

Write a MATLAB code for discrete least squares trigonometric polynomial S3(x), using m = 4 for f(x) = e^x * cos(2x) on the interval [-pi, pi]. Compute the error E(S3).

In: Advanced Math

1. Consider the following second-order differential equation. d^2x/dt^2 + 3 dx/dt + 2x − x^2 = 0 (a) Convert the equation into a first-order system in terms of x and v, where v = dx/dt. (b) Find all of the equilibrium points of the first-order system. (c) Make an accurate sketch of the direction field of the first-order system. (d) Make an accurate sketch of the phase portrait of the first-order system. (e) Briefly describe the behavior of the first-order system

In: Advanced Math

Cadbury is preparing a special edition of its famous
egg-fondant. The new design of the chocolate shell of the small egg
is created from the following two paraboloids: z =
2−2x^{2}−2y^{2} and z = x^{2} +
y^{2}−1. The x, y and z coordinates are given in
centimeters. **Answer the following questions using triple
integrals and show steps.**

a) Determine the volume of the egg using the **rectangular
coordinates**

b) Knowing that the density of the fondant inside the egg (in g
/ cm3) at a point (x, y, z) is 3 times the distance from this point
to the Oz axis, determine the mass of the egg using **the
cylindrical coordinates**. (We neglect here the mass of the
chocolate shell.)

c) You share this egg with your teacher by cutting it with the
horizontal plane z = c. Determine the value of the constant c to
separate your egg into two pieces of equal volumes.**Use
rectangular coordinates**

d) Determine the mass of each of the two pieces found in (c)
using the **cylindrical coordinates**. Which piece
should you keep to have the one with the most fondant?

*** fondant is the chocolate white milk inside the egg ****

In: Advanced Math

Qualitative,
Quantitative, Discrete, Continuous.

In: Advanced Math