A certain dynamical system is governed by the equation x^'' + (x')² + x = 0. Show that the origin is a center in the phase plane and that open and closed paths are separated by the path

2y² = 1- 2x

Find two linearly independent solutions near the regular singular point x₀= 0

x²y'' + (6x + x²)y' + xy = 0

I have a group order 28 is it possible to have an orbit with 6

maybe use sylos thm.

Question C [SD1: 5 Marks]

            A multiple regression analysis between yearly income (Y in $1,000s), college grade point average (X₁), age of the individuals (X₂), and the gender of the individual (X₃; zero representing female and one representing male) was performed on a sample of 10 people, and the following results were obtained.

In parts a and b, A is a matrix representation of a linear transformation in the standard basis. Find a matrix representation of the linear transformation in the new basis. show all steps.

a. A = 2x2 matrix with the first row being 2 and -1, and the second row being 1 and 3; new basis = {<1, 2> , < 1, 1> }

b. A = 3x3 matrix with the first row being 2, 1, -1, the second row being, 0, 1, 3, and the third row being -1, 2, 1. new basis = {< 0, -2, 1 > , <1, 2, 0> , <1, 1, 1,>}.

consider the subspace W=span[(4,-2,1)^T,(2,0,3)^T,(2,-4,-7)^T]

Find

A) basis of W

B) Dimension of W

C) is vector v=[0,-2,-5]^T contained in W? if yes espress as linear combantion

Solve the given system of differential equations by systematic elimination. (D + 1)x + (D − 1)y = 6 ; 7x + (D + 6)y = −1

How to determine all the basic solutions of an optimization problem, and classify them as feasible or infeasible?

As an example,

Maximize z= 2x₁ + 3x₂

Subject to:

x₁ + 3x₂ <= 6

3x₁ + 2x₂ <= 6

x₁ + x₂ >= 0

Some solutions are (2,0), (0,2), (0,0), (6/7, 12/7), but are there more solutions? And how do you tell if they are infeasile/feasible? Would you need to graph it?

Consider the set of vectors {(?,?,?) ∈ ?³???ℎ ?ℎ?? ? − 3? + 5? = 0.} Does this set of vectors form a sub-space of ?³?

Let R be the relation on Q defined by a/b R c/d iff ad=bc. Show that R is an equivalence relation. Describe the elements of the equivalence class of 2/3.

Solve the given initial-value problem. y''' − 2y'' + y' = 2 − 24ex + 40e5x, y(0) = 1 2 , y'(0) = 5 2 , y''(0) = − 5 2

Draw a graph having the given properties stated below, or explain why no such graph exists: In each case assume simple graphs (no self loops and no parallel edges) a. Six vertices each with degree 3 b. Five vertices each with degree 3. c. Four vertices each with degree 1. d. Six vertices and four edges. e. Four edges; four vertices having degrees 1, 2, 3, and 4.