Consider the second-order differential equation x2y′′+(x2+ax)y′−axy=0 where a=−2 Is x0=0 a singular or ordinary point of the equation? If it is singular, is it regular or irregular? Find two linearly independent power series solutions of the differential equation. For each solution, you can restrict it to the first four terms of the expansion

Solve ODE (3x - 2y + 1)dx + (-2x + y + 2)dy = 0 with the method of x =u + h and y = v + k

A teacher puts 10,000 lolly pops into a chest weighing 10 pounds and drops it down a well. A group of students are trying to get the lolly pops out the well. A student decided to attach to the chest a 100 ft long chain weighing 10 pounds. Several other students, at the top of the well, start lifting the chest out of the well by pulling on the chain in a hand over hand fashion. As the chest is being lifted, lolly pops fall from an opening in the chest at a constant rate such that the chest will be empty when it reaches the top of the well. Given that an individual lolly Pop weight 0.6 ounces, find the work done in raising the chest to the top of the well. Show integration steps.

ASSUME THE HYPERBOLIC PARALLEL POSTULATE. SHOW AS COMPLETE ARGUMENTS AS POSSIBLE AND INDICATE ALL STEPS OF REASONING. DIAGRAMS SHOULD ILLUSTRATE ALL ARGUMENTS, BUT REMEMBER THAT DIAGRAMS BY THEMSELVES DO NOT CONSTITUTE PROOFS.

Let l be a line and let P be a point not on l.

Prove that there exists a line m through P such that m is parallel to l but l and m do not admit a common perpendicular

(6) Define a binary operation ∗ on the set G = R^2 by (x, y) ∗ (x', y') = (x + x', y + y'e^x)

(a) Show that (G, ∗) is a group. Specifically, prove that the associative law holds, find the identity e, and find the inverse of (x, y) ∈ G.

(b) Show that the group G is not abelian.

(c). Show that the set H= (x*x=e) is a subgroup of G.

A professor gives two types of quizzes, objective and recall. He plans to give at least 15 quizzes this quarter. The student preparation time for an objective quiz is 15 minutes and for a recall quiz 30 minutes. The professor would like a student to spend at least 5 hours (300 minutes) preparing for these quizzes above and beyond the normal study time. The average score on an objective quiz is 7,and on a recall type 5, and the professor would like the student to score at least 85 points on all quizzes. It takes the professor one minute to grade an objective quiz, and 1.5 minutes to grade a recall type quiz. How many of each type should he give in order to minimize his grading time?

Find the inverse of the matrix

A=
2 -1 3

0 1 1

-1 -1 0

A mass of 1 slug, when attached to a spring, stretches it 2 feet and then comes to rest in the equilibrium position. Starting at t = 0, an external force equal to f(t) = 10 sin(4t) is applied to the system. Find the equation of motion if the surrounding medium offers a damping force that is numerically equal to 8 times the instantaneous velocity. (Use g = 32 ft/s² for the acceleration due to gravity.)

x(t) = ____________ft

​Explain why most folks say that alcohol concentration decays linearly until the alcohol concentration decays to the level at which no one cares.​

Vectors p1 = [10 2] and p2 = [2 15] are represented in o₁x₁y₁ coordinate frame. Translate o₁x₁y₁ by t = [30 1]^T and rotate by 45^◦ to obtain o₂x₂y₂. Is the dot product of p₁ and p₂ same in o₁x₁y₁ and o₂x₂y₂?

I believe o₁x₁y₁ and o₂x₂y₂ are the pair of vectors before and after the translation and rotation

How many ways can a gardener plant five different species of shrubs in a circle? What is the answer if two of the shrubs are the same? What is the answer if all the shrubs are identical?

Use separation of variables to solve uxx+2uy+uyy=0, u(x, 0) = f(x), u(x, 1) = 0, u(0, y) = 0, u(1, y) = 0.

Each short-answer question below can be answered with a few sentences.

2. A colleague of yours is faced with testing a relationship at the .05 level or significance or the .01 level of significance. What is your advice to your colleague? Why?

3. A colleague of yours has made a decision to decrease the sample size because of limited funding. What should they expect to happen to the probability of uncovering statistically significant findings in their research? Why?

4. What information is provided by calculations of the “chi-square” test statistic?

5. What information is provided by calculations of the “t” test statistic?