Use the Method of Undetermined Coefficients to find the general solution

1) y''-3y'+2=cos(x)

2) y''-3y'+2=xe^x

(a) Let G and G′ be finite groups whose orders have no common factors. Show that the only homomorphism φ:G→G′ is the trivial one.

(b) Give an example of a nontrivial homomorphism φ for the given groups, if an example exists. If no such homomorphism exists, explain why.

i.φ: Z16→Z7

ii.φ: S4→S5

Group of Symmetries of a Rectangle

a. Carefully describe the group of symmetries of a rectangle Describe the types, the orders, and the structures of the groups and their elements. After clearly naming the elements in some way, provide tables for each group. Describe them as a group of permutations on the vertices.

b. Next, carefully describe each of these groups as subgroups of some permutation group. Be sure to provide reasons for your choices.

c. What are the POSSIBLE orders for any subgroups of each group? Explain.

d. Next, carefully describe all the subgroups of each of these groups. Be sure to provide information about the structure of each subgroup, their order, the order of their elements. Provide generator(s) where possible.

e. Answer these questions about each group described in part a making sure to give reasons: Are any of these groups cyclic? Are any of these groups abelian? Which groups are cyclic? Which groups are abelian? Are there subgroups of every possible order? Which subgroups (in each group among the different groups) are isomorphic? How do you know they are isomorphic or not?

Group of Symmetries of a Cube

a. Carefully describe the group of symmetries of a cube. Describe the types, the orders, and the structures of the groups and their elements. After clearly naming the elements in some way, provide tables for each group. Describe them as a group of permutations on the vertices.

b. Next, carefully describe each of these groups as subgroups of some permutation group. Be sure to provide reasons for your choices.

c. What are the POSSIBLE orders for any subgroups of each group? Explain.

d. Next, carefully describe all the subgroups of each of these groups. Be sure to provide information about the structure of each subgroup, their order, the order of their elements. Provide generator(s) where possible.

e. Answer these questions about each group described in part a making sure to give reasons: Are any of these groups cyclic? Are any of these groups abelian? Which groups are cyclic? Which groups are abelian? Are there subgroups of every possible order? Which subgroups (in each group among the different groups) are isomorphic? How do you know they are isomorphic or not?

The quantity demanded x of a certain brand of DVD player is 3000/week when the unit price p is $485. For each decrease in unit price of $20 below $485, the quantity demanded increases by 250units. The suppliers will not market any DVD players if the unit price is $350 or lower. But at a unit price of $525, they are willing to make available 2500 units in the market. The supply equation is also known to be linear.

(a) Find the demand equation.
p(x) =  

(b) Find the supply equation.
p(x) =  

(c) Find the equilibrium quantity and the equilibrium price.

Which of the following statements are correct?

a) If A is a bounded subset of the real line, every infinite subset of A has a limit point.

b) If A is a bounded subset of the real line, every open cover of A has a finite subcover.

c) If A is an infinite open subset of the real line, there is an infinite open cover with a finite subcover.

d) If A is a closed subset of the real line, every open cover of A has a finite subcover.

Just chose the correct answer choice.

Given the nxn matrices A,B,C of real numbers, which satisfy the Condition:

A+B+λΑΒ=0

Β+C+λBC=0

A+C+λCA=0

for some λ≠0 ∈ R

(α) Prove that I+λΑ,Ι+λΒ,Ι+λC are invertible and AB=BC=CA.

(b) Prove that A=B=C

Translate the following argument into symbolic form and determine weather it's logically correct by constructing a truth table. Money causes all the world's troubles or money helps the poor. If money helps the poor, it is not the cause of all the worlds troubles. Money is the cause of all the world's troubles. Therefore, money does not help the poor. Please show how the problem was solved!

Prove that if U, V and W are vector spaces such that U and V are isomorphic and V and W are isomorphic, then U and W are isomorphic.

Solve the given system of differential equations by systematic elimination.

consider the following equation,

max r = 4x + y + 6z

2x + y + 2z <= 10

x + 2y + z <= 9

x + 2z <= 6

x, y, z >= 0

The tableau corresponds with a step of the SIMPLEX method applied to the previous problem

what is the value of x, y, z and the objective function. Identify the entering and the leaving variables. Justify your response with the calculations performed (you can extend the previous tableau to perform the required calculations)

The quantity demanded x each month of Russo Espresso Makers is 250 when the unit price p is $135. The quantity demanded each month is 1000 when the unit price is $105. The suppliers will market 700 espresso makers when the unit price is $66. At a unit price of $96, they are willing to market 2200 units. Both the supply and demand equations are known to be linear.

(a) Find the demand equation.
p =  

(b) Find the supply equation.  
p =  

(c) Find the equilibrium quantity and the equilibrium price.

Using Java (Swing) language(please hard code)... Create a program that has a textfield for the user to type in a set of input. Below that textfield have the following controls to show string manipulations:

(1) A button that will change the entire textfield’s current text to uppercase.

(2) A button with its own textfield for a search value, that will tell the position the search value appears in the textfield above.

(3) A button that reports the current number of characters in the textfield above.