Consider the following equation: (3 − x^2 )y'' − 3xy' − y = 0 Derive the general solution of the given differential equation about x = 0. Your answer should include a general formula for the coefficents.

If G = (V, E) is a graph and x ∈ V , let G \ x be the graph whose vertex set is V \ {x} and whose edges are those edges of G that don’t contain x.

Show that every connected finite graph G = (V, E) with at least two vertices has at least two vertices x1, x2 ∈ V such that G \ xi is connected.

1.

a) The demand per week for television sets is 1200 units when the price is $575 each and 800 units when the price is $725 each. Find the demand equation for the sets, assuming that it is linear?

b) Suppose a manufacturer of some product will produce 10 units when the price is SR150 each and 6 units when the price is SR70 each. Find the supply equation, assuming it is linear?

c) In each of the following, sketch the given function and find its slope:

f(x)=3x+1

f(x)-2x

Prove by contraposition and again by contradiction:

For all integers a,b, and c, if a divides b and a does not divide c then a does not divide b + c

Elaboration with definitions / properties used would be appreciated!

Thanks in advance!!

a) Your initial belief about stock A is that its future price cannot be predicted on the basis of existing public information. An insider comes forward claiming that the price will fall. You know the insider is not totally reliable and tells the truth with probability p=0.3. Use Bayes’ theorem to calculate the posterior probability that the stock price will fall, based on the insider’s evidence.A second insider, equally unreliable, comes forward and also claims that the price will fall. Assuming that the insiders are not colluding, what is your posterior probability of a price fall?  Based on your above answers, does the probability of future stock price depend on unreliable insiders? Would you expect this outcome? Explain your argument.

(9) (a) If f(t) = x₁e ^−kt cos(ωt) + x₂e ^−kt sin(ωt), then f'(t) has the same form with coefficients y₁, y₂.

(i) Find the matrix A such that y = Ax.

(ii) Find the characteristic polynomial of A.

(iii) What can you say about eigenvalues of A? (iv) Interpret your answer to

(iii) as a calculus statement. That is, explain how your answer to (iii) could have been predicted from a basic fact of calculus.

(b) If f(t) = x₁ sin(t) + x₂ cos(t) + x₃tsin(t) + x₄t cos(t), then f'(t) has the same form with coefficients y1, ..., y4. Same questions (a)-(d) as previously.

(i) Find the rate of change of the function f(x) = (x + 2)/( 1 − 8x) with respect to x when x = 1. (ii) The number of units Q of a particular commodity that will be produced with K thousand dollars of capital expenditure is modeled by Q(K) = 500 K ^(2/3). Suppose that capital expenditure varies with time in such a way that t months from now there will be K(t) thousand dollars of capital expenditure,where K(t) = (2t^4 + 3t + 149)/( t + 2) (a) What will be the capital expenditure 3 months from now? How many units will be produced at this time? (b) At what rate will production be changing with respect to time 5 months from now? Will production be increasing or decreasing at this time?

On the island of Knights and Knaves we have three people A, B and C. (The island must be known for its inhabitants’ very short names.) A says: We are all knaves. B says: Only one of us is a knave. Using an approach similar to the one in the notes, determine if A, B and C are each a knight or a knave. (The problem might have no solutions, one solution, or many solutions.)

The number of commercial airline boardings on domestic flights increased steadily during the 1990s as shown in the table below. Let​ f(t) be the number of commercial airline boardings on domestic flights​ (in millions) for the year that is t years since 1990.

Numbers of Commercial Airline Boardings on Domestic Flights

f f(x)

Year Number of Boardings​ (millions)

1991 452

1995 547

1997 599

1999 635

2000 666

Find an equation of f.

Let (X,d) be the Cartesian product of the two metric spaces (X_1,d₁) and (X_2,d₂).

a) show that a sequence {(x_n¹,x_n²)} in X is Cauchy sequence in X if and only if {x_n¹} is a Cauchy sequence in X₁ and {x_n²} is a Cauchy in X_2.

b) show that X is complete if and only if both X₁ and X₂​​​​​​​ are complete.

Now assume that the harvesting is not done at a constant rate, but rather at rates that vary at different times of the year. This can be modeled by ??/?? = .25? (1 − ?/4 ) − ?(1 + sin(?)). This equation cannot be solved by any technique we have learned. In fact, it cannot be solved analytically, but it can still be analyzed graphically.

8. Let c=0.16. Use MATLAB to graph a slopefield and approximate solutions for several different values of p(0), and interpret what you see. Turn in the graphs together with your analysis. Note: 0<p<5, and 0<t<50 is a reasonable viewing window to start with, though you may want to change it as you proceed.

a) Verify that the indicial equation of Bessel's equation of order p is (r-p)(r+p)=0

b) Suppose that p is not an integer. Carry out the computation to obtain the solutions y1 and y2 above.

y"+y=cos(9t/10)

1. general solution of corresponding homongenous equation

2. particular solution

3.solution of initial value problem with initial conditions y(0)=y'(0)=0

4. sketch solution in part 3