($4.7 Cauchy-Euler Equations): Solve the following Euler-type equations (a)–(c).

(a) x^2y''-4xy'-6y=0

(b) x^2y''+7xy'+13y=0

(c) x^2y''+3xy'+y=x

($4.6 Variation of Parameters): Solve the equations (a)–(c) using method of variation of parameters.

(a) y''-6y+9y=8xe^3x

(b) y''-2y'+2y=e^x (secx)

(c) y''-2y'+y= (e^x)/x

Assume there are three subsets X, Y, Z of some universal set U.

| X u Y u Z | = 41
| X | = 20
| Y | = 28
| Z | = 21

| X n Y | = 12
| X n Z | = 10
|Y n Z | = 11
|X-| = 24 (bar over top of X).

Solve:

a) | X u Y |

b) | Y △ Z |

c) | X n Y n Z |

d) |Y - (X u Z) |

e) | U |

f) |X n Y n Z | (one bar over both X and Y).

Define a sequence from R as follows. Fix r > 1. Let a1 = 1 and define recursively, an+1 = (1/r) (an + r + 1). Show, by induction, that (an) is increasing and bounded above by (r+1)/(r−1) . Does the sequence converge?

Please solve the following:

u_t=u_xx, 0<x<1, t>0

u(0,t)=0, u(1,t)=A, t>0

u(x,0)=cosx, 0<x<1

if s₁ and s₂ are two simple functions then prove that the max and minimum of then are also simple function.

If |G| = p1* p2* p3 (distinct primes)
Then G has a normal subgroup.

Give a proof, base the proof on the Determinant of a Vandermonde matrix that the INTERPOLATING POLYNOMIAL exist and its unique.

Provide the definition of an Empirical Model, and provide two examples.

*** PLEASE EXPLAIN ANSWER ***

Dear Students:

I have recently been employed by HMS Nautical Inc to work on their submarine program. I have only some basic data to work with and no idea how to use it to get the information I need.

Here is what I know. First, I know that our Subs have a maximum running depth of 500 feet below sea level. I also know that a sub functioning at acceptable levels should be able to reach maximum depth in 10 minutes. Finally I know that I need to multiply the decent by a factor of 5 to achieve an accurate model. I also have a chart that lists times and depths for the sub.

Finally, I know that the sub follows a quadratic model when it descends and then ascends.

I have been told that you will be able to take this data and make sense of it. I would like a model for the path of the submarine as it descends to its running depth and then returns to the surface. I want to be able to use this model to predict where the sub will be at any time during its decent/ascent cycle. I would also like to know after how many minutes I should expect the sub to breech the surface of the water again.

Please explain clearly how you came up with the model so that I can repeat the process for new additions to our fleet of submarines. I appreciate any help you can give me in this matter. I would like your response returned to me either as a business letter !!!

Sincerely,

Nemo Hook

HMS Nautical In

Prove the converse of Theorem 3.3.4 by showing that if a set K ⊆ R is closed and bounded, then it is compact.

Theorem 3.3.4 A set K ⊆ R is compact if and only if it is closed and bounded.

Solve the given differential equation by undetermined coefficients. y'' − 8y' + 20y = 200x2 − 65xex

1. Let ρ: R2 ×R2 →R be given by ρ((x1,y1),(x2,y2)) = |x1 −x2|+|y1 −y2|.

(a) Prove that (R2,ρ) is a metric space.

(b) In (R2,ρ), sketch the open ball with center (0,0) and radius 1. 2. Let {xn} be a sequence in a metric space (X,ρ). Prove that if xn → a and xn → b for some a,b ∈ X, then a = b.

3. (Optional) Let (C[a,b],ρ) be the metric space discussed in example 10.6 on page 344 of Wade. If {fn} and f are in C[a,b], prove that fn → f with respect to ρ if and only if fn → f uniformly (in the sense of Chapter 7).

Prove the following.

Let T denote the integers divisible by three. Find a bijection f : Z→T (Z denotes all integers).