Consider the ODE y"+ 4 y'+ 4 y = 5 e^(− 2 x ). (

a) Verify that y 1 ( x) = e − 2 x and y 2 ( x) = xe − 2 x satisfy the corresponding homogeneous equation.

(b) Use the Superposition Principle, with appropriate coefficients, to state the general solution y h ( x ) of the corresponding homogeneous equation.

(c) Verify that y p ( x) = 52 x 2 e − 2 x is a particular solution to the given nonhomogeneous ODE.

(d) Use the Nonhomogeneous Principle to write the general solution y ( x ) to the nonhomogeneous ODE.

(e) Solve the IVP consisting of the nonhomogeneous ODE and the initial conditions y(0) = 1 , y 0 (0) = − 1 .

In: Advanced Math

Describe a problem that you are currently faced with at work or in your personal life that could be solved by using an optimization model. Describe what the problem is, why optimization modeling could help you and how you would approach solving the problem with an optimization model.

In: Advanced Math

a) If you add the terms ax^2+bxy+cy^2 to the function L(x,y) = 1 - 0.5y, how would this affect the derivatives of L(x,y) at (0,0)?

b) What a, b and c you would pick to make those derivatives match the derivatives of the f(x,y) = sqrt(x^2+1-xy-y) at (0,0).

c) Define this new L(x,y).

In: Advanced Math

For this activity, select a recurring quantity from your OWN
life for which you have monthly records at least 2 years (including
24 observation in dataset at least). This might be the cost of a
utility bill, the number of cell phone minutes used, or even your
income. If you do not have access to such records, use the internet
to find similar data, such as average monthly housing prices, rent
prices in your area for at least 2 years (You must note the data
source with an accessible link). Data can also be monthly sales of
some particular commodity. 1.4 Please do the descriptive analysis,
using the method of index number and Exponential Smoothing
individually. And try to explain the pattern you find. 1.5 Use two
methods you learned to predict the value of your quantity for the
next year (12 months). And make comparison with two results.

In: Advanced Math

Find the best weights (w0...w4) of the highest possible order finite difference formula of the form f'(x) ~ w0*f(x) + w1*f(x+h) + w2*f(x+2h) + w3*f(x+3h) + w4*f(x+4h) and use Taylor series to predict the convergence order as h is decreased.

In: Advanced Math

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for every n ∈ N. That is, the sum of the first n perfect cubes is the square of the sum of the first n natural numbers. (As a student, I found it very surprising that the sum of the first n perfect cubes was always a perfect square at all.)

In: Advanced Math

7. Let m be a fixed positive integer.

(a) Prove that no two among the integers 0, 1, 2, . . . , m − 1 are congruent to each other modulo m.

(b) Prove that every integer is congruent modulo m to one of 0, 1, 2, . . . , m − 1.

In: Advanced Math

The following is a payoff table giving costs for various situations. What decision would an optimist make?

State 1 | State 2 | State 3 | |

Alternative 1 | 45 | 37 | 83 |

Alternative 2 | 16 | 59 | 72 |

Alternative 3 | 23 | 65 | 91 |

Alternative 4 | 44 | 33 | 55 |

In: Advanced Math

Let Σ *⊆* *P rop*(*A*). Show that
Σ*|−* *p* iff Σ *∪ {¬**p**}* is
unsatisfifiable.

In: Advanced Math

Using concepts from funding models, how can they refocus its activities to generate funds to continue the work of the organization?

In: Advanced Math

I'm having trouble figuring out the constraints to this problem. I know that I am maximizing 55x + 45y, however the variables are throwing me off. I only need help on question #1 as it would be a great help to understanding the rest of what the questions are asking. The problem is as follows:

NorCal Outfitters manufactures a variety of specialty gear for outdoor enthusiasts. NorCal has decided

to begin production on two new models of crampons: the Denali and Cascade. The company produces

crampons by first stamping steel sheets into the rough design, then assembling the base crampon with

toe and ankle straps. NorCal’s manufacturing plant has 120 hours of stamping time and 80 hours of

assembly time assigned for producing these crampons.

Each set of Denali crampons requires 30 minutes of stamping time and 25 minutes of assembly time,

and each set of Cascade crampons requires 25 minutes of stamping time and 15 minutes of assembly

time. The labor and material cost is $15 and $10 for each set of Denali and Cascade crampons,

respectively. NorCal sells crampons through wholesale distributors for $55 for the Denali model and $45

for the Cascade model. The V.P of Production at NorCal believes that the Denali model, recently

featured in Outside Magazine, could become a bestseller and has determined that at least 60% of the

crampons produced by NorCal should be the Denali model.

1. Solve this problem using the graphic solution technique.

In: Advanced Math

There are eight different kittens at a store. I have three different children - how many ways are there to give each child a different kitten?

In: Advanced Math

At time t = 0 a tank contains 25lb of salt dissolved in 100 gallons of water. assume that water containing 2lb salt/gallon enters the tank at a rate of 5 gal/min and the well-stirred solution is leaving the tank at the same rate.

solve for Q(t) [Amount of salt in tank at time t ]

In: Advanced Math

3. To begin a proof by contradiction for “If n is even then n+1 is odd,” what would you “assume true?

4. Prove that the following is **not** true by
finding a **counterexample**.

“*The sum of any 3 consecutive integers is
even"*

5. Show a **Proof by exhaustion** for the
following: *For n = 2, 4, 6, n²-1 is
odd*

** 6. **Show an informal

**Recursive Definitions**

** 7. **The Fibonacci Sequence
is defined as
follows:

*F(1) = 1*

*
F(2) = 1*

*
F(n) = F(n-1) + F(n-2) for n>2.*

The first 10 numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

Find F(13) = ______________ and F(14) =______________

8. Prove the following property of the Fibonacci numbers directly from the definition.

*F(n+3) = 2F(n+1) + F(n-3)+ 2F(n-2)*

9. Find the 3^{rd}, 4^{th} and
5^{th} values in the sequence.

*D(1) = 0*

*
D(n) = 2D(n-1) + 2 for n>1.*

D(2) = _______________________________________________

D(3) = _______________________________________________

D(4) = _______________________________________________

10. Find the 3^{rd}, 4^{th} and
5^{th} values in the sequence.

*D(1) = 1*

*
D(2) = 4*

*
D(n) = 2D(n-1) + D(n-2) for n>2.*

D(3) = _______________________________________________

D(4) = _______________________________________________

D(5) = _______________________________________________

11. Using the SelectionSort algorithm on p 169 in your textbook, simulate the execution of the algorithm on the following list L; write the list after every exchange.

**L: 9, 2, 4, 8, 6**

_______________

_______________

_______________

_______________

In: Advanced Math

**Harvesting Fish.** A fish farmer has 5000 catfish
in a pond. The number of catfish increases by 8% per month and the
farmer harvests 420 catfish per month.

a) **(4 points)** Find a recursive equation for the
catfish population Pn for each month.

b) **(4 points)** Solve the recursive equation to
find an explicit equation for the catfish population.

c) **(3 points)** How many catfish are in the pond
after six months?

d) **(3 points)** Is harvesting 420 catfish per
month a sustainable strategy for the fish farmer?

e) **(4 points)** What is the maximum number of
catfish the fish farmer can harvest to sustain his business?

In: Advanced Math