5. Equations of the form y’ = P(x)*y^2 + Q(x)*y + R(x) are called Riccati equations.

i) If we know a solution y = φ(x) of this equation, then any other solution

can be written in the form y(x) = φ(x)+ 1/v(x), where v(x) is an unknown

function which satisfies a certain linear equation. Using the fact that

φ and y both solve the above Riccati equation, find the differential

equation that v satisfies.

ii) Consider the equation 3y’ + y^2 +2/(x^2) = 0. Find one solution of this equation by inspection.

iii) Use the method of part(i) to find the general solution of the equation

in (ii).

Given the following utility matrix, representing the ratings, on a 1–5 star scale, of eight items, a through h, by three users A, B, and C:

After we normalize the matrix by subtracting from each nonblank entry the average value for its user, what is the cosine distance between users A and C ?

One particle travels along the path

p1(t) = <2.666 cos(6.405t + 5.149) + 4.430, 2.666 sin(6.405t + 5.149) − 3.610, 11.18t + 6.633>

and another along the path

p2(t) = <1.084t + 3.125, 3.096t − 5.332, −2.925t + 4.377>.

The paths intersect at two points, one of which is a collision. Find the point where the particles collide and the other point where the paths intersect.

Jennifer is the owner of a video game and entertainment software retail store. She is currently planning to retire in 30 years and wishes to withdraw $11,000/month for 20 years from her retirement account starting at that time. How much must she contribute each month for 30 years into a retirement account earning interest at the rate of 2%/year compounded monthly to meet her retirement goal? (Round your answer to the nearest cent.)

$________

Assignment 1 - IN PDF FORMAT Using R and Rstudio

Pick a database from: data()

Then preview the first 10 rows.

Print the number of rows and columns

- Print the names of the variables

If you have row names, print them - work with the values for a field in your dataset. You can do it by dataset[[xx]] operator with xx can be the index of the field or the nae of the field.

Now use dataset[xx] to get a slice of datframe instead of vector, e.g. df = mtcars[c(‘mpg’, ‘disp’)] - With two scalar arguments (rows,columns), the [] operator can select elements in the dataframe. Try this.

- Subset your data using rules similar to the lectures. For example part of the data that has a characteristics AND/OR another one, etc.

- Write a function that does an operation, for example averaging a property, on your dataset.

- Write a function that works on rows and adds a column to the dataset with the results for each row.

- Using the iris database, explain the following line of code:

data(iris) aggregate(iris["Petal.Length"], by = iris["Species"], FUN=median)

## Species Petal.Length

## 1 setosa 1.50

## 2 versicolor 4.35

## 3 virginica 5.55

For extra credit for my Differential Equation class, I have to come up with a real-world example of a system of linear differential equations (three) such as a population, spring, or circuit and solve it and then write a paper about it. I can write the paper no problem but I need help coming up the equation and then solving it, the equation should be roughly 8-11 steps.

For the example given to us, (WHICH WE CANNOT USE) the person put a resistor circuit into a system of linear equations and then defined ohms law as eq(1) I=V/R then stated the sum as eq(2) then put the ohms and resistors into wronskin determinant and solved for sq(3)&(4) and provided a figure for each or every other one which I should be able to come up with if you cannot.

I need help coming up with the idea and the first couple sets of equations, I can do the research, I just need an idea and an system of equations presented and solved

Expressions and Equations

Think about a scenario when you would use an expression and an equation in a real life application. Discuss each scenario and explain how to write the expression and equation. What is the difference between both of them?

Create two sample equations for your peers to solve.

Please give me an answer ASAP Something new that not already on the Chegg site.

a) Show that if A is a real, non-singular nxn matrix, then A.(A^T) is positive definite.

b) Let H be a real, symmetric nxn matrix. Show that H is positive definite if and only if its eigenvalues are positive.

How are Histograms and Pareto Charts in Excel created? Give an example using sample dataset.

How can I code this problem in MATLAB:

a) Find the approximations to within 10^-4 to all real zeros of the following polynomials using Newton's method.?

f(x)=x³ - 2*x²- 5.

b) Find approximations to within 10^-5 to all the zeros of each of the following polynomials by first
finding the real zeros using Newton’s method and then reducing to polynomials of lower degree to
determine any complex zeros.

f(x)=x⁴ + 5x³ - 9*x² - 85*x - 136.

Solve STEP BY STEP using power series about x = -1,

4(x+1)^2 y'' - 2(x+1)(x+3) y' + (x+4) y = 0.

note, solve by power series, variable transformation by letting:
w= x+1 = 0

Instructions: Search the news, or online articles and sites to find the term “probability” (likelihood, odds, and expected outcome are also acceptable terms rather than “probability”). Locate two examples, one where you believe it is used correctly and one that you think seems fishy. Post your sources as a reference. No need to explain the articles, or how the term is used because that will be done in our response later in this module.

Assume you have been recently hired by the Department of Transportation (DoT) to analyze motorized vehicle traffic flows. Your initial goal is to analyze the traffic and traffic delays in a large metropolitan area.

You choose to use a weighted graph to represent this particular scenario. Remember that a graph is a collection of nodes (or vertices) and edges. Each edge will have a corresponding weight.

Describe how you would construct such a weighted graph. What do the nodes represent? What do the edges represent? What values would the weights represent? Is this graph a tree? Justify your design. Feel free to include an example in your description.

Describe an adjacency matrix you might create to represent this graph. What do the rows represent? What do the columns represent? What do the entries in the matrix represent? Feel free to create an example or create an illustration to better explain yourself.

Suppose you wish to identify the shortest travel time between nodes on the graph. What type of path would represent such a route? Describe this path in detail.

Suppose that the DoT is considering repairing some of the roads in the metropolitan area. But first, the DoT wishes to assess each and every road to determine which of the roads require repair. You have reported to the manager of the repair project and noted that you can use your graphical representation and graph-based methods to suggest the most efficient route the inspection crew can take to inspect all the roads. Describe in detail and name the type of the particular path on the graph that would represent this most efficient route. Consider all aspects of travel time undertaken by the inspection crew. Justify your answer.

Suppose that the DoT is planning to widen roads that create a bottleneck in the flow of traffic. As such, you pose this problem as a max flow/min cut problem. You have determined that roads are in need of widening if the traffic flow on the roads is at maximum capacity; these roads are essentially the bottlenecks of the road system. These bottleneck roads can be identified using the Max Flow/Min Cut theorem.

Using the graph representation of the road system, explain how the Max Flow/Min Cut theorem can be used to identify the bottleneck roads.

Create an example graph one might use to analyze this network as a Max Flow problem. Include at least 8 nodes and 16 edges. Include capacities and flows and identify the max flow and/or min cut.

Write a brief report on a real-life project that can be organized using PERT. Books on operations research and project management often have sections on PERT which is sometimes called the critical path method or CPM. Find references on the internet for commercial software that draws PERT diagrams and describe sine of these products and their applications.

compare and contrast slope and average rate of change

identify the characteristics of the graph of a quadratic equation

give a real life application of a quadratic function