Let G(x,t) be the Green's function satisfying

          G"(x,t) + G(x,t) = δ(x-t)

with G(0,t) = 0 and G(pi/2) = 0, and the neccessary continuity conditions.

Here t is fixed, 0 < t < pi/2; the derivative means differentiation with respect to x, 0 < x < pi/2

Fill in the blanck:

G(x,t)=                                                   when 0<x<t

                                                             when t<x<pi/2

A department contains 10 part-time employees and 15 full-time employees. In how many ways can a committee be formed with 6 members if it must have more full-time employees than part-time employees?

a) Find the right form of the particular solution (you don't have to solve it)

y''-6y'+9y=6x^2+2-12e^3x

b) Solve using variation of parameters

y" + y = secx

c) define or explain the following

- Linear system

- Linear Homogenous system

- Linear Homogenous system with Constant coefficients

d) Solve the initial value problem using the D elimination

dx/dt = 4x - 3y

dy/dt = 6x-7y

Subject to the I.C

x(0) = 2

y(0) = -1

a) Check e^5x, xe^5x for independence using Wronskian

b) solve y" + 5y' + 6y = 0

c) convert the above equation to a system and solve it

d) Solve x^2y''-2xy'-4y= 0 by assuming solution of the form y=x^r

e) Solve the IVP using undetermined coefficients

y" -2y' -3y = 6

y(0)= 3

y'(0)=11

Use Lisp to achieve an informed search(8-queen): 8 queens randomly placed on a board and returns a list that consists of the same queens in locations that they don't threaten each other. Requirement : the input configurations. the number of nodes expanded and the final solution

Let G = Z4 ⊕ Z4, and H = {(0, 0), (2, 0), (0, 2), (2, 2)}, and K = (1, 2). Is G/H isomorphic to Z4 or Z2 ⊕ Z2? Is G/K isomorphic to Z4 or Z2 ⊕ Z2?

You are interested in studying the relationship between a home’s selling price and various features of the home (this is called a hedonic price analysis). You estimate the following regression equation, in thousands of dollars:
  
(house price) ̂=175+12bedroom+.8square_footage - 2.5distance

Where bedroom measures the number of bedrooms in the house, square_footage is the house’s square footage, measured in hundreds (e.g. 2,200 square foot house would enter the regression as 22), and distance is the distance from downtown, measured in miles.

a. Do the signs of the coefficients match your a priori expectations? Explain why or why not based on your intuition.

b. Suppose you are a realtor. Use the regression results to estimate the value of a client’s home with the following characteristics: 3,200 square feet; 4 bedrooms; located 6 miles from downtown.
  
c. How would your estimate change if the client decided to add a 200 square-foot bedroom?

1. Express the degree of a vertex in terms of the adjacency matrix
2. Describe how you can find out whether a graph is connected, based on its adjacency matrix, using only matrix addition and multiplication.

Without using the Tutte-Berge formula, prove that every cubic graph with at most two bridges contains a perfect matching.

Predict the form of the solution (study limits) and find all of the solutions using the Frobenius method approach. Write indicial equation, find its roots, the recurrence relation, and the first four terms terms of each series solution for xy"+y'+x^2y=0

proof of the general stokes theorem using the greens and divergence theorem that is understandable by a calculus student

find the general solution of the following as follows
Xn+2 = -2Xn+1 + 3Xn, x0=1 x1=2

a) find the 2x2 matrix that satisfies Yn+1=AYn
b) Find the characteristic value of A and its corresponding characteristic vector
c) express X0 = (1 2)as a linear combination of characteristic vector
d) find Yn
e) find Xn

Roberts Company, a small machine shop, is contemplating acquiring a new machine that costs
$24,000. Arrangements can be made to lease or purchase the machine. The firm is in the 40%
tax bracket. The firm would obtain a 5-year lease requiring annual end-of-year lease payments
of $6,000. All maintenance costs would be paid by the lessor, and insurance and other costs
would be borne by the lessee. The lessee would exercise its option to purchase the machine for
$1,200 at termination of the lease.
The firm would finance the purchase of the machine with a 9%, 5-year loan requiring end-of-
year installment payments of $6,170. The machine would be depreciated by 20% in year 1, 32%
in year 2, 19% in year 3, and 12% in years 4 and 5. The firm would pay $1,500 per year for a
service contract that covers all maintenance costs; insurance and other costs would be borne by
the firm. The firm plans to keep the machine and use it beyond its 5-year recovery period.

i. Find the after-tax cash outflows for each year under the lease alternative.
ii. Find the after-tax cash outflows for each year under the purchase alternative.
iii. Calculate the present value of the cash outflows associated with the lease and purchase
alternatives using the after-tax cost of debt as the discount rate.
iv. Choose the alternative with the lower present value of cash outflows from Step 3. It will
be the least-cost financing alternative.

For each given integer n, determine if n is a sum of two squares. (You do not have to find the squares.)

(a)n= 19

(b)n= 45

(c)n= 99

(d)n= 999

(e)n=1000

Find the infinite series for the following differential equation about x = 0, using Frobenius method, Bessel's or Legrende's equations.

x^2y" + 4xy' + (2+x)y = 0