Use Laplace Transforms to solve the following IVPs . y′′+9y={2−t ,0} piecewise function 0≤t<2 , t≥2 ;y(0)=1 ,y′(0)=0

PART A)

Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 5300. The number of fish doubled in the first year.

Assuming that the size of the fish population satisfies the logistic equation

dPdt=kP(1−PK),

determine the constant k, and then solve the equation to find an expression for the size of the population after t years.
k=_______________  
P(t)=______________

How long will it take for the population to increase to 2650 (half of the carrying capacity)?
It will take ________________ years.

PART B)

Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation

dPdt=cln(KP)P

where c is a constant and K is the carrying capacity.

Solve this differential equation for c=0.2, K=3000 and initial population P0=200
P(t)=__________ .

Compute the limiting value of the size of the population.
limt→∞P(t)=_________

At what value of PP does PP grow fastest? ___________
P=__________

Q1. In a second-semester English class, 35% of the students are sophomores and the rest freshmen; 30% of the sophomores are repeating the course and 20% of the freshmen gained admission to the course by advanced placement. The professor randomly selects a student to comment on the assignment. (Round your answers to three decimal places.)

• Find the probability that the professor selects a freshman who gained advanced placement admission to the class.?

A) In a group of college students, 60 males and 80 females, 25% of the males and 45% of the females are from out of state. A student is randomly selected. Find the following probabilities. (See Example 9. Round your answers to three decimal places.)

• The person is a female.?
• The person is from out of state, given that the person is a male.?
• The person is from out of state when the person is selected from the females.?
• The person is from out of state.?
• The person is female when the selection is made from the out-of-state persons.?
  

A hardware store will run an advertising campaign using radio and newspaper. Every dollar spent on radio advertising will reach 60 people in the "Under $35,000" market, and 60 people in the "Over $35,000" market. Every dollar spent on newspaper advertising will reach 100 people in the "Under $35,000" market, and 20 people in the "Over $35,000" market. If the store wants to reach at least 210,000 people in the "Under $35,000" market and 240,000 people in the "Over $35,000" market, how much should it spend on each type of advertising to minimize the cost?

Minimum amount spent on advertising (in dollars):

Dollars spent on radio advertising

Dollars spent on newspaper advertising

Use Newton's method to find all solutions of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)

sqrt(x + 1) = x^2 − x

What does x equal?

Let A be a real n × n matrix, and suppose that every leading principal submatrix ofA of order k < n is nonsingular. Show that A has an LU-factorisation.

sketch the fourier series of f(x) on the interval -L <= x <= L for

A) f(x) = (x when x < L/2 and 0 when > L/2)

B) f(x) = e^-x

Prove that the set R ={ a+ b√2+c√3+d√6 , a,b,c,d belongs to Q } is a field

Use Octave

Given a matrix M ∈ M mn , each function should first ensure that the matrix has the proper size (e.g., be
square if the definition involves a square matrix). Until CA4, we do not implement proper error handling,
so for now, if the matrix is not of the proper size, the function should return FALSE. In this assignment,
we create functions that characterise matrix properties or types. These definitions can be found easily. For
reference, the location of some of the definitions in the Horn & Johnson book is provided. The following
functions should be made available:  
1. is_ real _matrix(M) is true if M ∈ M(R).
2. is _complex _matrix(M) is true if M ∈ M(C) and ∃i,j such that =(m ij ) 6= 0.
3. is _diagonal_ matrix(M) is true if M ∈ M(C) is a diagonal matrix (H&J 0.9.1).
4. is_ lower _triangular_ matrix(M) is true if M ∈ M(C) is a lower triangular matrix (H&J 0.9.3).
5. is _upper _triangular_ matrix(M) is true if M ∈ M(C) is an upper triangular matrix (H&J 0.9.3).
6. is _triangular _matrix(M) is true if M ∈ M(C) is a triangular matrix (H&J 0.9.3).
7. is _symmetric_ matrix(M) is true if M ∈ M(C) is a symmetric matrix.
8. is -hermitian _matrix(M) is true if M ∈ M(C) is a Hermitian matrix.
9. is_ skew _hermitian _matrix(M) is true if M ∈ M(C) is a skew Hermitian matrix.
10. is_ normal _matrix(M) is true if M ∈ M(C) is a normal matrix.
11. is _unitary_ matrix(M) is true if M ∈ M(C) is a unitary matrix.
12. is_ orthogonal_ matrix(M) is true if M ∈ M(C) is an orthogonal matrix.
13. is_ permutation _matrix(M) is true if M ∈ M(C) is a permutation matrix (H&J 0.9.5).
14. is _reversal_ matrix(M) is true if M ∈ M(C) is a reversal matrix (H&J 0.9.5).
15. is _circulant _matrix(M) is true if M ∈ M(C) is a circulant matrix (H&J 0.9.6).
16. is _Toeplitz_ matrix(M) is true if M ∈ M(C) is a Toeplitz matrix (H&J 0.9.7).
17. is _Hankel _matrix(M) is true if M ∈ M(C) is a Hankel matrix (H&J 0.9.8).
18. is _lower_ Hessenberg matrix(M) is true if M ∈ M(C) is a lower Hessenberg matrix (H&J 0.9.9).
19. is_ upper_Hessenberg matrix(M) is true if M ∈ M(C) is an upper Hessenberg matrix (H&J 0.9.9).
20. is _tridiagonal_ matrix(M) is true if M ∈ M(C) is a tridiagonal matrix (H&J 0.9.10).
21. is _Jacobi _matrix(M) is true if M ∈ M(C) is a Jacobi matrix (H&J 0.9.10).
22. is_ persymmetric_ matrix(M) is true if M ∈ M(C) is a persymmetric matrix (H&J 0.9.10).

why is the slant height s = sqrt(2) - x . Please explain thoroughly how to find the slant height.

(Folding a pyramid) A pyramid with a square base and four faces, each in the shape of an isosceles triangle, is made by cutting away four triangles from 2 ft square piece of cardboard and bending up the resulting triangles to form the walls of the pyramid. What is the largest volume the pyramid can have? Hint: The volume of a pyramid having base area A and height h measured perpendicular to the base is V = (1/3)Ah.

Solve using matlab code

The initial value problem dydx−y= 2 cosx, y(0) =−2

has the exact solution y(x) =−e^x −√2 cos (x+π4).

Use the Euler method to solve the initial value problem for 0≤x≤2 using n=10,50,100,200 and plot solutions in one graph.

Repeat #1 using the Runge-Kutta method and plot solutions in one graph with the exact solution

In biotechnology, where do you use differential equations? What kind of problems can we solve with them?

The Johnsons have accumulated a nest egg of $40,000 that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of $2300/month in monthly payments (to take advantage of the tax deduction) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed $2900. If local mortgage rates are 3.5%/year compounded monthly for a conventional 30-year mortgage, what is the price range of houses that they should consider? (Round your answers to the nearest cent.)

How to proof:

Matrix A have a size of m×n, and the rank is r. How can we rigorous proof that the dimension of column space are always equal to the dimensional of row space?

(I can use many examples to show this work, but how to proof rigorously?)

x² y" + (x²+x) y’ +(2x-1) y = 0,

1. Find the general solution of y₁ with r₁ and calculate the coefficient up to c₄ and also find the general expression of the recursion formula, (recursion formula for y₁)
2. Find the general solution of y₂ based on theorem 4.3.1. (Hint, set d₂ = 0)