3 Solve the following IVP

a) y"+8y'-9y=0 y(1)=1, y'(1)=0

b)y"+4y'+5y=0 y(0)=1, y'(0)=0

c) y"-6y'+9y=0 y(0)=0, y'(0)=2

d)y"+4y'+4y=0 y(-1)=2, y'(-1)=1

If you add the terms ax^2+bxy+cy^2 to the function L(x,y) = 1 - 0.5y, how would this affect the derivatives of L at (0,0)? What a,b and c you would pick to make those derivatives match the derivatives of the f(x,y) = sqrt(x**2+1-x*y-y) at (0,0). Define this new L(x,y).

Following the Well Ordering Principle for Integers suppose n is an arbitrary integer greater than 1 and:

Z={x is a positive integer such that x is greater than or equal to 2 and x divides n}

Let y be the least element is set Z. Using proof by contradiction, show that y is prime.

Let X, Y, Z be sets. Find a bijection (with proof) between X × (Y × Z) and X × Y × Z.

Kindly request to solve the Poisson's equation of Uxx+Uyy = -81xy ; 0<x<1, 0<y<1; given that u(0,y)=0, u(x,0)=0; u(1,y)=100, u(x,1)=100 and h=1/3

Use the formula for continuous compounding to compute the balance in the account after​ 1, 5, and 20 years.​ Also, find the APY for the account.

A ​$6000 deposit in an account with an APR of 3.83.8​%.

The balance in the account after 1 year is approximately ​$

​(Round to the nearest cent as​ needed.)

The balance in the account after 55 years is approximately ​$

​(Round to the nearest cent as​ needed.)

The balance in the account after 20 years is approximately ​$

​(Round to the nearest cent as​ needed.)

The APY for the account is approximately ​%

A1-C1

Assignment – Correlation

1. You are given the following statistics: means, standard deviations, and correlations for 4 variables. The four variables are miles per gallon, weight of car, front profile (1-Low, 0-Not Low) and the manufacturer’s suggested retail price. Four hundred cars were sampled. Answer the questions that follow.

1. __________Which costs more, (low or high) front profile vehicles
2. _________ Best predictor of MPG is
3. __________Heavier cars have a (low or high) profile front
4. __________High MPG vehicles have ‘significantly’ higher prices than low MPG vehicles(T/F)
5. _________ Weakest predictor of MPG is
6. __________Slope of the line for MPG against Low Profile Front would be (Pos, Neg or zero)
7. ____________________________________________________________ Suppose the slope of the line for MPG against Low Profile front is 5. Interpret this slope relative to the two variables.

h.__________Higher MPG vehicles have (higher or lower) profiles.

i.__________ How many cars in the sample were low profile.

eCampusOntario.ca is the primary face of the Ontario Online Learning Consortium (OOLC), a not-for-profit corporation whose membership is composed of all publicly-funded colleges and universities in Ontario. Funding for OOLC and the eCampusOntario initiative comes from the Government of Ontario. eCampusOntario.ca was created in 2015 primarily as a portal for learners to find online courses – through browsing, searching by keyword, or filtering by institution and delivery format. Course details include links to information about scheduling, tuition and fees, instructors, and information about how to get registered and to seek credit transfer. Previously identified course equivalencies are also listed where available. In addition, students can search based on a specific course they need credit for to find available online options at other institutions. Aiming to improve its services, eCampusOntario has selected a sample of 33 nursing, 33 business and 33 education students who intend to use eCampusOntario website to search and register for e-courses. The sampled students were asked to indicate their overall satisfaction of eCampusOntario portal prior to registration for any courses (stage 1) and after registration (stage 2). The following information is available for each respondent: - Age of the respondents - Gender (1=male and 2=female) - Student satisfaction of stage 1 (a continuous number that ranges from 0 (extremely dissatisfied) to 5 (extremely satisfied)) - Student satisfaction of stage 2 (a continuous number that ranges from 0 (extremely dissatisfied) to 5 (extremely satisfied)) * Assume the collected data is normally-distributed.

Assignment Questions: 1. Conduct a descriptive analysis of the sample of business stream students by choosing two variables and focusing on measures of central location and variance. Represent these variables using two appropriate graphs for each. In addition, represent the relationship between age and student satisfaction for each stage using an appropriate graph. (15%)

2. Some eCampusOntario employees believe more information is provided to business students during stage 1 (pre registration) as opposed to stage 2 (post registration). Is there evidence to suggest there are significant differences between the business student’s satisfactions of the two stages. (30%)

3. Estimate the difference between business student satisfaction during stage 1 and 2 with 95% confidence. Interpret the results. (20%)

4. eCampusOntario would also like to know if differences exist between the satisfaction level of nursing students, business students and education students during stage 2. Assume the variables have equal variance. (30%) Assume alpha = 0.05 wherever necessary. Hint: Pay attention to the experimental design; are the samples independent or paired?

Answer each question using Excel. Do each question on a separate Excel Sheet.

Answer each question using Excel. Do each question on a separate Excel Sheet. - For #2 – 4 do the following: • State your chosen method of analysis and justify its use. Were the required conditions met? If the question involves testing a hypothesis, state the hypotheses. Include the Excel results in readable format. • Discuss the findings of the study by answering the following questions: ➢ What are they? ➢ What do they mean? ➢ How can they help in decision making?

Consider the backward heat equation ∂tu + uxx = 0 with u(0, x) = g(x) in the periodic domain x ∈ T (Peridoc) . Use separation of variables to solve this equation and prove that as long as g is not a constant, the solution does not decay whent → ∞.

Solve the given differential equation by (a) undetermined coefficients and (b) variation of parameters:

y'' -3y'+2y=sinx

Solve the following initial value problems by the method of undetermined coefficients:

A.) y′′ + y = 2 + sin t.

B.) y′′ + 4y′ + 5y = e−2t(1 − cos t).

Create the generating functions in closed form ( not as an infinite sum) using the sequences:

example: 1,1,1,1,1,1,... = 1/(1-x)

1. 0,1,1,1,1,1,...

2. 1,0,0,1,0,0,1,0,0,1,0,0,1,...

3. 0, 0,0,0,1,1,1,1,1,1,1,1,1,...

4. 1, -1, 1, -1, 1,....

5. 1,1,1,0,0,1,1,1,1,1,1,1,1,1,...

6. 0,0,0,1,2,3,4,5,6,..

7. 3,2,4,1,1,1,1,1,...

8. 0,2,0,0,2,0,0,2,0,0,..

Consider the method of steepest descent for the function f(x, y) = x^2 − y^2 . Note: this is a quadratic function, but the matrix Q is not positive definite.

(a) Find a formula for α_k. (b) For which initial points (x0, y0) is (x1, y1) = (x0, y0)?

For any sequence (x_n) of real numbers, we say that (x_n) is increasing iff for all n, m ∈ N, if n < m, then x_n < x_m. Prove that any increasing sequence that is not Cauchy must be unbounded. (Here, “unbounded” just means that x_n eventually gets larger than any given real number). Then, show that any increasing sequence that is bounded must converge

Let R and S be rings. If R is isomorphic to S, show that R[x] is isomorphic to S[x].