y'''-2y"-y'+2y=xe^{x^2+2x}

a) Find a general solution to the corresponding homogeneous equation, given that e^2x is one.

b) In the method of variation of parameters, find v₁, where v₁e^2x + v₂y₂ + v₃y₃ = y_p is a particular solution to the inhomogeneous equation. Use the method of variation of parameters

Please explain and show work, thanks!

Define f : Z × Z → Z by f(x, y) = x + 2y. (abstract algebra)

(a) Prove that f is a homomorphism.

(b) Find the kernel of f.

Farmer Pickles wants Bob to paint the circular fence which encloses his sunflower field. If the parametric equations x = 18 cos(θ) and y = 18 sin(θ) describe the base of the fence (in yards) and the height of the fence is given by the equation h(x, y) = 12 + (2x − y)/6, then how many gallons of paint with Bob need to complete the project. Assume that one gallon of paint covers three hundred square feet of fence

(3) Use the Hungarian Method to find the minimum possible cost for assigning the following jobs to the following four workers. The three jobs are new flooring, a new roof, and a new boiler. The costs for each person are as follows: (25 pts.)

Steve: 105 for Flooring, 321 for Roofing, 580 for Boiler.

Hoshi: 215 for Flooring, 300 for Roofing, 500 for Boiler.

Iqbal: 150 for Flooring, 315 for Roofing, 520 for Boiler.

Daphne: 240 for Flooring, 280 for Roofing, 497 for Boiler.

Recall that a 5-bit string is a bit strings of length 5, and a bit string of weight 3, say, is one with exactly three 1’s.

a. How many 5-bit strings are there?

b. How many 5-bit strings have weight 0?

c. How many 5-bit strings have weight 1?

d. How many 5-bit strings have weight 2?

e. How many 5-bit strings have weight 4?

f. How many 5-bit strings have weight 5?

g. How many 5-bit strings have weight 9?

Bob signs a note promising to pay Marie $3875 in 3 years at 9.5% compounded monthly. Then, 102 days before the note is due, Marie sells the note to a bank which discounts the note based on a bank discount rate of 18.5%. How much did the bank pay Marie for the note? $

Assume that you are breaking a stick into 3 pieces by uniformly and independently selecting two break points ? and ?. If we denote the event that these pieces form a triangle by T and lengths of the three pieces by ?, ? and ?, calculate: a. ?(? ∩ (? < ? < ?)), b. The probability that the pieces form an equilateral triangle.

Bezout’s Theorem and the Fundamental Theorem of Arithmetic

1. Let a, b, c ∈ Z. Prove that c = ma + nb for some m, n ∈ Z if and only if gcd(a, b)|c.

2. Prove that if c|ab and gcd(a, c) = 1, then c|b.

3. Prove that for all a, b ∈ Z not both zero, gcd(a, b) = 1 if and only if a and b have no prime factors in common.

Let S_k(n) = 1^k + 2^k + ··· + n^k for n, k ≥ 1. Then, S_4(n) is given by

S_4(n)= n(n+1)(2n+1)(3n^2 +3n−1)/ 30

Prove by mathematical induction.

Adam is loaning $5000 to Bert for a period of 2 years. Suppose Bert will repay the loan with a $5000 balloon payment at the end of the 2 years and will pay monthly interest payments each month which will end the month before the $5000 balloon payment. e. If interest is 6% effective interest, how much will Bert’s interest payments be to Adam each month? f. If Adam takes the payments he receives each month from Bert and reinvests them at 3% annual interest compounded monthly, how much will he have in his savings account at the time that Bert pays the balloon payment? g. What is Adam’s rate of return on his initial investment of $5000? That is find the yield rate as an effective interest rate.

Part II: Choose the best answer

7. Which of the following is a fundamental product adjacent to xy'zw
   1. xyzw'
   2. xy'zw'
   3. x'y'z'w
   4. x'yzw
   5. None
8. The quantity of fundamental products when you have 3 Boolean variables is:
   1. 4
   2. 6
   3. 15
   4. 8
   5. None
9. Which of the following is not a fundamental product adjacent to xy'zw
   1. xy'z'w
   2. x'yz'w
   3. x'y'zw
   4. xyzw
   5. None
10. Morgan's 1st law is defined (x + y) '= x'y' How do you simplify (x + y + z + w)' using this law?
    1. x'+y'+z'+w'
    2. (x+y)'(z+w)'
    3. x'y'z'w'
    4. (xy)'+(zw)'
    5. None
11. Morgan's 2nd law is defined (xy)'= x' + y ' How do you simplify (xyz)' using this law?
    1. x'+y'+z'
    2. x'+y'z''
    3. x'y'z'
    4. x'y'+z'
    5. None
12. Use Morgan's 1st and 2nd law, to simplify
    [(w + x) y] '
    1. w'+x'+y'
    2. w'+x'y'
    3. w'x'y'
    4. w'x'+y'
    5. None
13. Use Morgan's 1st and 2nd law to simplify [(x + y)'z']' Remember that (x')'= x
    1. xyz
    2. (x+y)z
    3. x+y+z
    4. (xy)+z
    5. None
14. Use Morgan's 1st and 2nd law, to simplify
    [(w + x + y) z] '
    1. w'x'y'z'
    2. w'+x'+y'+z'
    3. (w'x'y')+z'
    4. (w'+x'+y')z'
    5. None
15. The quantity of fundamental products when there are n Boolean variables is
    1. n+2
    2. 2n
    3. n²
    4. 2ⁿ
    5. None

find Lagrange polynomials that approximate f(x)=x^3,

a) find the linear interpolation p1(x) using the nodes X0=-1 and X1=0

b) find the quadratic interpolation polynomial p2(x) using the nodes x0=-1,x1=0, x2=1

c) find the cubic interpolation polynomials p3(x) using the nodes x0=-1, x1=0 , x2=1 and x3=2.

d) find the linear interpolation polynomial p1(x) using the nodes x0=1 and x1=2

e) find the quadratic interpolation polynomial p2(x) using the nodes x0=0 ,x1=1 and x2=2

Find the product from the multiplication table for the symmetries of an equilateral triangle the new permutation notation to verify each of the following:

A) R(240)R3 B) R3R(240) C) R1R3 D) R3R1