Use the Laplace transform to solve the given initial value problem.

y'' + 2y' + 10y = 6cos2t - 4sin2t, y(0)=2, y'(0)= -2

how much do wild mountain lions weigh? adult wild mountain lions captured and released for the first time in the san Andres mountains gave the following weights 68 108 125 125 60 64

For which real values of a do there exist solutions of the differential equation

y'' + 2y' + ay = 0

which satisfy the conditions y(0) = y(π) = 0 but which are not identically zero? For each such a give an appropriate non-zero solution

A. Find a particular solution to the nonhomogeneous differential equation y′′ + 4y′ + 5y = −15x + e^-x

y =

B. Find a particular solution to

y′′ + 4y = 16sin(2t).

y_p =

C. Find y as a function of x if

y′′′ − 10y′′ + 16y′ = 21e^x,

y(0) = 15,  y′(0) = 28, y′′(0) = 17.
y(x) =

Question1: Consider a QR faction M=QR, show that R= Transpose(Q)M

You need to show that (1)M = QR where R := Transpose(Q)M and (2) that R is upper triangular.

To show (1) use the fact that QTranspose(Q) is the matrix for orthogonal projection onto the image of M. What happens to a column of M (which is a vector in the image of M) when you project it onto the image of M?

To show (2), think about the entries of R := Transpose(Q)M as dot products between the columns v_1,...,v_n of M and the rows u_1,...,u_n of Q^T. Entries of Transpose(Q)M vanish when these vectors are orthogonal. The vectors u_1,..., u_n are the othonomal basis for the image of M obtained from v_1,...,v_n via the Gram-Schmidt process. Why is it the case that u_i.v_j =0 if i>j?

(c) [2] For which of the following functions are the level curves linear?  

(I) f(x, y) = tan(x + y)

(II) g(x, y) = e^y/x (e to the power of y over x)

(III) h(x, y) = ln(xy)

(A) none (B) I only (E) I and II (F) I and III

(C) II only (G) II and III

(D) III only (H) all three

A partial table of values for a function f(x,y) is given below. Which of the following are positive?

(I) fy(4, 1)

(II) fx(4, 1) (III) fxx(4, 1)

Y''+y'-20y=xe^3x+e^4x

Find the general solution of this differential equation

why the cylindrical shape of a can is the optimal shape

Needs to be 2 pages.

Solve the following linear integer programming model using the Cutting Plane method. Show all relevant work in your solution report.

Maximize Z = x1 + x2

Subject to

3x1 + 2x2 < 5

x2 < 2

x1, x2 > 0 and integer.

Show that every permutational product of a finite amalgam am(A,B: H) is finite.Hence show that every finite amalgam of two groups is embeddable in a finite group.

A country's census lists the population of the country as 254 million in 1990, 286 million in 2000, and 314 million in 2010. Fit a second-degree polynomial passing through these three points. (Let the year 2000 be x = 0 and let p(x) represent the population in millions.)

p(x) = ________million

Use this polynomial to predict the populations in 2020 and in 2030.

2020_________million

2030 ________million

Show the following identities for a, b, c ∈ N.

(a) gcd(ca, cb) = c gcd(a, b) Hint: To show that two integers x, y ∈ Z are equal you can show that both x | y and y | x which implies x = y or x = −y. Thus, if both x and y have the same sign, they must be equal.

(b) lcm(ca, cb) = c lcm(a, b)

(c) ab = lcm(a, b) gcd(a, b) Hint: Consider first the case that gcd(a, b) = 1 and show that ab = lcm(a, b) in this case. For the general case combine this with (b).

(d) lcm(gcd(a, c), gcd(b, c)) = gcd(lcm(a, b), c) Hint: First treat the special case that gcd(a, b, c) = 1. In this case begin by showing that lcm(gcd(a, c), gcd(b, c)) = gcd(a, c) gcd(b, c). The asserted equality gcd(a, c) gcd(b, c) = gcd(lcm(a, b), c) is then shown by proving that gcd(a, c) gcd(b, c)| gcd(lcm(a, b), c) and gcd(lcm(a, b), c)| gcd(a, c) gcd(b, c). Proceed to show that gcd(a, c)| lcm(a, b) and gcd(a, c)| c, and deduce from this that gcd(a, c)| gcd(lcm(a, b), c); proceed analogously for gcd(b, c). Then argue that gcd(a, c) gcd(b, c)| gcd(lcm(a, b), c) under the present assumption. Conversely, in order to show that gcd(lcm(a, b), c)| gcd(a, c) gcd(b, c), write according to (a) gcd(a, c) gcd(b, c) = gcd(gcd(a, c)b, gcd(a, c)c) = gcd(gcd(ab, bc), gcd(ac, c2 )), and show that gcd(lcm(a, b), c) divides all of ab, bc, ac, and c 2 . Explain from here why gcd(lcm(a, b), c) must divide gcd(a, c) gcd(b, c) then as well. For the general case explain how (a) and (b) can be used to reduce the general assertion to the previously treated special case.

***The only help I really need is with c and d. I just added a and b for context.

Here are four questions:

1. Prove a standard Brownian motion is Gaussian process .

2. Prove a Brownian bridge is Gaussian process.

3. Prove Ornstein-Uhlenbeck process is Gaussian

4. Prove the position process is Gaussian.

Please provide as many detail as you can, thanks.