Find two linearly independent power series solutions of the given differential equation about the ordinary point x=0.

y''-2xy=0

Consider the equation u_tt = u_xx

x ∈ (0, pi)

u_x(0,t) = u(pi,t) = 0

Write the series expansion for a solution u(x,t)

The price of a home in Medford was $ 100,000 in 1985 and rose to $ ⁢ 170 ,000 in 1999 .
a. Create two models, f ( t ) assuming linear growth and     g ( t ) assuming exponential growth, where t is the number of years after 1985 . Round coefficients to three decimal places when necessary. f ( t ) =     g ( t ) =

b. Fill in the following table using the equations you just found. Use the exact equation for exponential growth.
Round your answers to the nearest integer.

1) A radioactive substance decays at a rate proportional to the amount of the substance at present time. Initially 200 grams of a the substance was present and remain 80% of the initial amount after 2 hours.

A.) Determine the amount of the substance remaining after 10 hours (counted in grams)

B.) Determine the time that 60% of the initial amount of the substance has decayed (counted in hours)

Use the following statement in the subsequent parts of the question: “If Ann is Jan’s mother, then

Jose is Jan’s cousin.”

a)

Write the converse, inverse, and contrapositive of the statement.

b)

Show that the statement and the contrapositive are equivalent.

c)

Show that the converse and inverse are equivalent.

d)

Based on your findings, is the inverse equivalent to the contrapositive? Justify your answer.

consider the following LP

a. put the problem into standard form, using slack, excess, and artificial variables

b. solve the associated LP of phase I

c. is w'=0, proceed to phase 2 and solve the original LP. show all of the work.

max z=x1-x2+3x3

S.T. x1+x2<=20

x2+x3>=10

x1+x3=5

x1,x2>=0

In June 2001 the retail price of a 25-kilogram bag of cornmeal was $8 in Zambia; by December the price had risen to $11.† The result was that one retailer reported a drop in sales from 14 bags per day to 2 bags per day. Assume that the retailer is prepared to sell 4 bags per day at $8 and 16 bags per day at $11. Find linear demand and supply equations, and then compute the retailer's equilibrium price.

Let T : Rn →Rm be a linear transformation.

(a) If {v1,v2,...,vk} is a linearly dependent subset of Rn, prove that {T(v1),T(v2),...,T(vk)} is a linearly dependent subset of Rm.

(b) Suppose the kernel of T is {0}. (Recall that the kernel of a linear transformation T : Rn → Rm is the set of all x ∈ Rn such that T(x) = 0). If {w1,w2,...,wp} is a linearly independent subset of Rn, then show that {T(w1),T(w2),...,T(wp)} is a linearly independent subset of Rm.

Find u(x,y) harmonic in S with given boundary values:

1. S = {(x,y): 1 < y < 3} , u(x,y) = 5 (if y=1) and = 7 (when y=3)
2. S = {(x,y): 1 < x² + y² < 4}, u(x,y)= 5 (on outer circle) and = 7 (on inner circle)

I have these two problems to solve, and I'm not sure where to start. Any help would be appreciated. Thanks!

1. (a) Construct a magic Venn diagram (MVD) or order 3, i.e., label the eight non-overlapping regions of a Venn diagram of 3 generic sets A, B, C with the integers 1 through 8 so that the sum of the labels in each of the three sets is a constant m (called the 'magic sum' of the diagram.

1. (b) Construct another MVD of order 3 with a magic sum different from the one you found in part (a).

1. (c) Construct an MVD of order 4 (which has 4 sets A, B, C, D, and 16 labels).

If a sound with frequency fs is produced by a source traveling along a line with speed vs. If an observer is traveling with speed vo along the same line from the opposite direction toward the source, then the frequency of the sound heard by the observer is fo = c + vo c − vs fs where c is the speed of sound, about 332 m/s. (This is the Doppler effect.) Suppose that, at a particular moment, you are in a train traveling at 30 m/s and accelerating at 1.5 m/s2. A train is approaching you from the opposite direction on the other track at 32 m/s, accelerating at 1.8 m/s2, and sounds its whistle, which has a frequency of 458 Hz. At that instant, what is the perceived frequency that you hear? (Round your answer to one decimal place.) Hz

How fast is it changing? (Round your answer to two decimal places.) Hz/s

e Elixer Drug Company produces a drug from two ingredients. Each ingredient contains the same three antibiotics, in different proportions. One gram of ingredient 1 contributes 3 units, and 1 gram of ingredient 2 contributes 1 unit of antibiotic 1; the drug requires 6 units. At least 4 units of antibiotic 2 are required, and the ingredients contribute 1 unit each per gram. At least 12 units of antibiotic 3 are required; a gram of ingredient 1 contributes 2 units, and a gram of ingredient 2 contributes 6 units. The cost for a gram of ingredient 1 is $80, and the cost for a gram of ingredient 2 is $50. The company wants to formulate a linear programming model to determine the number of grams of each ingredient that must go into the drug to meet the antibiotic requirements at the minimum cost.

1. Formulate a linear programming model for this problem.

2. make a table

3. Find critical points & evaluate the critical points

4. find the regions

5. Solve this model by using graphical analysis

Convert the complex number to polar form rcisθ.

(a) For z=−1+i
the modulus of zis r=, and the principal argument is θ =.

(b) For z=−33‾√−3i
the modulus of zis r=, and the principal argument is θ =.

(c) For z=−2
the modulus of zis r=, and the principal argument is θ =.

(d) For z=−3i
the modulus of zis r=, and the principal argument is θ= .