Exercise 4.9.29: Solve the following systems of congruences, or state that there is no solution. Be sure to state if there are multiple solutions.

a. {6 = 13a + b(mod 26), 13 = 4a + b(mod 26)

b. {14 = 17a + b(mod 26), 8 = 7a + b(mod 26)

c. {1 = 15a + b(mod 26), 10 = 9a + b(mod 26)

Goofus and Gallant try out for the quiz team. Goofus, Gallant, and seven of their classmates are trying out for the quiz bowl team. There are four positions available on the team.

(a) If each position is different, how many ways are there that the team can be made?
(b) If each position is the same, how many ways are there that the team can be made?
(c) Suppose each position is the same. How many teams include both Goofus and Gallant?
(d) Suppose each position is the same. How many teams include Goofus or Gallant? ("Or" means "at least one of.")

Solve listed initial value problems by using the Laplace Transform:

6.       y^ll − 3y^l − 4y = 4t − 5,     y(0) = 2, y^l(0) = 4

Find at least the first six nonzero terms in the power series expansion about x=0 for a general solution to the differential equation:

y'' + (sinx)y = 0

Given that x =0 is a regular singular point of the given differential equation, show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x = 0.

xy"+(1-x)y'-y=0

what is social contract and stakeholder theories of Corporate Social Responsibility (CSR) andexpalin use the social demandingness, social activist and stakeholder theories of corporate social responsibility in order to discuss the case study below.

Solve the differential equation

y''(x)-2xy'(x)+2ny(x)=0

using the Hermite Polynomials

Find two distinct subgroups of order 2 of the group D3 of symmetries of an equilateral triangle. Explain why this fact alone shows that D3 is not a cynic group.

For an integer k, define f(k) = gcd(11k + 1, 7k + 3).
(a) Compute R = {f(k): k ∈ Z}.
(b) For each n ∈ R, find a set D_n such that, for every integer k, f(k) = n if and only if k ∈ D_n.

Is there any solution without using the 'mod' for b?

2. Find the volume of revolution by WASHER: ? = 2?^(1/2) ??? ? = x