The solution to the Initial value problem x′′+2x′+17x=2cos(6t),x(0)=0,x′(0)=0 is the sum of the steady periodic solution xsp and the transient solution xtr. Find both xsp and xtr.

Abstract algebra:

Deduce the main theorem of Galois theory from Artin's lemma.

1. Let V = { S, A, B, a, b, λ} and T = { a, b }, Find the languages generated by the grammar G = ( V, T, S, P } when the set of productions consists of:
   1. S → AB, A → aba, B → bab.
   2. S → AB, S → bA, A → bb, B → aa.
   3. S → AB, S → AA, A → Ab, A → a, B → b.
   4. S → A, S → B, A → abA, A → a, B → baB, B → b.
   5. S → AB, A → aBb, B → bAa, A → λ, B → λ.

Consider the mathematical system with the single element {1} under the operation of multiplication. Answer and explain the following:

a)      Is the system closed?

b)      Does the system have an identity element?

c)      Does 1 have an inverse?

d)     Does the associative property hold?

e)      Does the commutative property hold?

f)       Is {1} under the operation of multiplication a commutative group?

The selling prices of all stocks listed on the CDNX stock exchange are known to be normally distributed with a mean of $20 and a standard deviation of $4. (a) What percentage of stocks have a selling price between $20 and $40? (b) If 167 of the stocks sell for $26 or higher, how many stocks sell on the CDNX exchange? (c) What is the minimum selling price of the most expensive 5% of stocks? (d) 2 out of every 3 stocks sells for higher than what price? (e) The prices of the middle 80% of stocks are between what two values?

There exists a group G of order 8 having the following presentation: G=〈i,j,k | ij=k, jk=I, ki=j, i^2 =j^2 =k^2〉. Denotei2 bym. Showthat every element of G can be written in the form e, i, j, k, m, mi, mj, mk, and hence that these are precisely the distinct elements of G. Furthermore, write out the multiplication table for G (really, this should be going on while you do the first part of the problem).

Suppose we had defined some bijection f : N → Q +.

(a) Discuss how you could use f to prove that Q − is countably infinite. That is, define a new function h : N → Q − that uses f in some way. Discuss why using f makes h itself a bijection.

(b) Discuss how you could show that Q is countably infinite.

1. Find the volume of the solid using triple integrals. The solid bounded below by the cone

z= sqr x2+y2 and bounded above by the sphere x2+y2+z2=8.(Figure)

1. Find and sketch the region of integration R.
2. Setup the triple integral in Cartesian coordinates.
3. Setup the triple integral in Spherical coordinates.
4. Setup the triple integral in Cylindrical coordinates.
5. Evaluate the triple integral in Cylindrical coordinates.

Hessian Matrix Determinant and Convex/Concave

I have a function which has a Hessian Matrix of:

Can anyone explain why this is neither a positive nor a negative definite?

And could you please explain 2*2 and 3*3 Hessian's rule of determining positive/negative definite? Thank you.

Let Z denote the set of all integers. Give an explicit bijection f : Z → N

Can someone please explain to me the relationship between Stoke's theorem, Green's theorem, and the Gradient theorem. and give one example each for 0,1 and 2 forms. As a Calc 3 student, I am very confused about what this theorem is, I know it is essential to what linear algebra. Please help.

Use Warshall’s algorithm to find the transitive closure of the relation R={(1,1),(1,3),(1,4),(2,4),(2,5),(3,1),(4,4),(5,3)}

prove that positive operators have unique positive square root

use Simpson's 3/8th Rule on the first 3 segments, and multiple application of Simpson's 1/3rd Rule on the rest of the segments.

?(?)=400?⁵−900?⁴+675?³−200?²+25?+0.2

a = 0.12
b = 1.56
n = 7

(a) Seek power series solutions of the given differential equation about the given point x0;
find the recurrence relation.
(b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates
sooner).
(c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set
of solutions.
(d) If possible, find the general term in each solution.

1. y''-y=0, x0=0

2. y''-xy'-y=0, x0=0

3. (4-x^2)y''+2y=0, x0=0

4. 2y''+(x+1)y'+3y=0, x0=2