Set up the appropriate form of the particular solution to each of the differential equations below, but do NOT determine the values of the coefficients.

(a) y′′ +10y′ +25y=2e^(5t) +te^(−5t)

b) y′′ +9y=5t^2 +4cos(3t)+6e^(3t)

10. Let G = D3 x Z2 x Z3. Let N = { (e,0,0), (d2,0,0), (e,1,0,), (d2,1,0) }. Find G/N .

*D3 is dihedral group 3 and d2 is diagonal flip in D3

Let G be a group of order 42 = 2 * 3 * 7

(a) Let P7 be a Sylow 7-subgroup of G and let P3 be a Sylow 3-subgroup of G . What are the orders of P3 and P7?

(b) Prove that P7 is the unique Sylow 7-subgroup of G and that P7 is normal.

(c) Prove that P3P7 is a subgroup of G

(d) Prove that P3P7 is a normal subgroup of G .

(e) Let P2 be a Sylow 2-subgroup of G . Prove that G \cong (P3P7) \Join P2

(f) Assume subgroup not abelian. WHat is the index of N G(P3) in G ? [G: N G(P3) ] = _______

You are a company and just issued a $20m 3-year fixed rate bond. Your investment bank suggests that you enter into an interest rate swap to turn this into a floating rate bond. Briefly discuss:

1. Whether the swap should be one where your company pays fixed and receives floating or the other way around.
2. Why your company might choose to enter the swap.
3. Why your company chose to issue a fixed rate bond rather than a floating rate bond.
4. Another choice that your company could take rather than using the interest rate swap

You should answer this question in a few short paragraphs.

let p1(x) = x^2-3x-10 ,p2(x)=x^2-5x+1,p3(x)=x^2+2x+3 and p4(x)=x+5

a- Using standard polynomIAL ADDITION ,what polynomial ax^2+bx+c can be expressed as linear combination of p1(x),p2(x),p3(X),p4(x)

b- a polynomial is equal to zero if and only if all it's coefficient to zero . solve for a1,a2,a3,a4 by expanding ,written as polynomial in x,and setting each coefficient equal to zero:

a1p1(x)+a2p2+a3p3(x)+a4p4(x)=0

Prove that a subgroup H of a group G is normal if and only if gHg−1 =H for all g∈G

A factorial experiment was designed to test for any significant differences in the time needed to perform English to foreign language translations with two computerized language translators. Because the type of language translated was also considered a significant factor, translations were made with both systems for three different languages: Spanish, French, and German. Use the following data for translation time in hours.

Test for any significant differences due to language translator system (Factor A), type of language (Factor B), and interaction. Use a =.05.

We want to design a rectangular box without a lid with a volume of 64000 cm^3. Find the dimensions that maximize the surface area. Using Lagrange Multipliers
Can you explain your reasoning pls

Calculate (1234/9887) and (4321/9887) as quadratic congruences.

1.(a) Show that the length of the broken line satisfies Length(L) ≥ |AB|.

(b) Show that L achieves the lower bound
Length(L) = |AB|

if and only if the vertices V1,...,Vk−1 all lie on the segment AB and appear in that orderonAB,i.e.,theysatisfyVi ∈Vi−1Vi+1 forall1≤i≤k−1.

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

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

y'' + 2y' − 8y = 4e^−3x − e^−x

1. A six-sided die is weighted so that all even numbers have an equal chance of coming up when the die is rolled, all odd numbers have an equal chance of coming up, and each even number is three times as likely to come up as each odd number. This die is rolled once.

What is the probability of rolling a 3?

What is the probability of rolling a 6?

What is the probability of rolling an even number?

2. An eight-sided die is weighted so that all even numbers have an equal chance of coming up when the die is rolled, all odd numbers have an equal chance of coming up, and each odd number is eight times as likely to come up as each even number. This die is rolled once.

What is the probability of rolling the number 5?
What is the probability of rolling the number 4?
What is the probability of rolling a number greater than 5?
Enter each answer as a whole number or a fraction in lowest terms.

3.A gumball machine contains 45 gumballs. Some are purple and the rest are yellow. There are 4 times as many purple gumballs as yellow. Because the purple gumballs are slightly smaller than the yellow, each purple gumball is 3 times as likely to be dispensed as each yellow gumball.
An experiment consists of the machine dispensing one gumball. Let each gumball be considered one outcome.

What weight should be assigned to each purple gumball?
What is the probability of the event that a yellow gumball is dispensed?
Enter your answers as whole numbers or fractions in lowest terms.

2. Find the general solution to the differential equation x^2y''+ y'+y = 0 using the Method of Frobenius and power series techniques.

1. Find the :

(a) Inverse using explicit Gauss-Jordan elimination

(b) The eigenvalues

(c) Respective eigenvectors

(2 0 -2

0 4 0

-2 0 5)

Determine whether the polynomials form a basis for P3:

1 − 2t ^2 , t + 2t^3 , 1 − t + 2t^2