State and sketch a proof of Cauchy’s Theorem (not Cauchy’s Integral Formula). You need not show all of the details, just describe the general steps.

Calculate the integral from (-infinty) to (+ infinity) of [x^2 / (x^4 + 4)[ dx

The half-life of carbon-14 is 5730 years. Assume that the decay rate is proportional to the amount. Find the age of a sample in which 10% of C-14 originally present have decayed.
Select the correct answer.

Part 4:

• Discuss in one paragraph which aspects of your roller coaster would realistically make sense in the real world and which aspects would not. Make sure you refer to graphs you used in your answer.
• Looking at the example below, explain why your function roller coaster cannot have a loop.

a) State Mellin’s inverse Laplace transform formula.
b) State Cauchy’s residue theorem.
iii. Use (a) and (b) to prove that the inverse Laplace transform of F(s)=1/(s+a) is equal to f(t)= e^(-at),t>0

Find the solution of the following problems. Before doing these problems, you might want to review Exercise 3** on page 63:

d.) xy" + y' = x, where y(1) = 1m and y'(1) = -1 (answer should be y(x) = 1/4 x² - 3/2 ln(x) + 3/4)

e.) (x-1)²y" + (x-1)y' - y = 0, where y(2) = 1, and y'(2) = 0 (answer should be: y(x) = 1/2 (x-1)^-1 + x/2 - 1/2)

**Exercise 3: The formula for a particular solution given in (3.42) applies to the more general problem of solving y" + p(t)y' + q(t)y = f(t). In this case, y₁ and y₂ are independent solutions of the associated homogeneous equation y" + p(t)y' + q(t)y = 0.

Please show work!

True or False.

1. If the set {v1,v2} is a basis of R^2, then the set {v1,v1+v2} is also a basis of R^2.

2.If W be a vector space and V1,V2 are subspaces of W, then V1 u V2 is also a subspace of W. V1 u V2 denotes the union of V1 and V2, i.e. the set of vectors which belong to either V1 or V2 (or to both).

3.If W be a vector space and V1,V2 are subspaces of W, then V1 ^ V2 is also a subspace of W. V1 ^ V2 denotes the intersection of V1 and V2, i.e. the set of vectors which belong to both V1 and V2.

Please explain why it is true and if it is false give a counterexample.

Find the six nonisomorphic trees on 6 vertices, and
for each compute the number of distinct spanning trees in K6 isomorphic
to it.

You and your classmates are boarding a lifeboat from a ship sinking in the middle of the ocean. You each are guaranteed a seat. Unfortunately, there are 10 others seeking a seat, as well, with only three seats remaining. As a group, you must decide which three people from the following list you will bring in your lifeboat and why?

• Restaurant manager, 67, married with two kids
• Croatian first-year medical student, 23, speaks little English
• Single mother, 34, has 3 kids
• LGBT male accountant, 58, a professional triathlete
• High school grad, 18, going into the army
• Homeless person, 40, Harvard dropout
• Successful entrepreneur, 49, chronic medical problems
• Top-ranked Olympic swimmer from Brazil, 21, no English
• Billionaire's daughter, 6, certified genius
• Comedian of Latino descent, 38, internationally famous

Define multiplication on ℤ? by ? ⊙ ? = ? where ? is the remainder when the product ?? is divided by ?.

(a) Show that this operation is commutative.

(b) Show that this operation is associative.

(c) Show that there is an identity element for this operation.

(d) Show that for ? = 5, the set of non-zero elements forms a group under multiplication.

(e) Show that for ? = 6, the set of non-zero elements does not form under multiplication.

Use the method of variation of parameters to find a particular solution of the given differential equation and then find the general solution of the ODE.

y'' + y = tan(t)

If T : R 3 → R 3 is projection onto a line through the origin, describe geometrically the eigenvalues and eigenvectors of T.

In each of the following cases, give an explicit formula for s_2m.

(1) s_2n = 2s_n+s_n, s1 = 0

(2) s_2n = 2s_n-n, s1 = 3

(3) s_2n = 2s_n+5-7n, s1 = 0

Differential Equations

22. Solve each of the following systems of equations.

(c) (D-2)x=0; -3x+(D+2)y=0; -2y+(D-3)z=0

I got x=c₁e^2t, y= 3/4 C₁e^2t + C₂e^-2t by using diffieq method d/dx(ye^2t) = 3C₁e^2t....., but the right answer is

x=4c₁e^2t, y= 3C₁e^2t + 5C₂e^-2t , z= -6C₁e^2t -2C₂e^-2t+C3e^3t

I want to know where 4 came from, and please do not use matrix system.