Solve the following DEs

1.) dy/dx + y =e^3x

2.) x^2y' +xy = 1

3,) x(dy/dx)-y = x^2 sin(x)

4.) y' - 2y = e^2x, y(0) = 2

5.) cos(x)(dy/dx) + ysin(x) = 1, y(pi/4) = 0

How many distinct arrangements of the letters in MANANGATANG are there in which the first two letters include an M or a T (or both)?

Note:

- MANANGATANG has 11 letters (one M, one T, two Gs, three Ns and four As).

- These are all the possible positionings of T and M in the first two positions:

PLEASE SHOW FULL LEGIBLE WORKING.

Use a graph or level curves or both to find the local maximum and minimum values and saddle points of the function. Then use calculus to find these values precisely. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = sin(x) + sin(y) + sin(x + y) + 8,    0 ≤ x ≤ 2π,    0 ≤ y ≤ 2π

Let U ⊂ C be open. Show that the conformal automorphisms of U form a group when the operation is taken to be composition.

Differential Equations with Matlab 3rd Edition Matlab Problem Set F Number 8:

Consider the predator-prey model dx/dt = x(4-3y) dy/dt = y(x-2) in which x >= 0 represents the population of the prey and y >= 0 represents the population of the predators.

a) Find all critical points of the system. At each critical point, calculate the corresponding linear system and find the eigenvalues of the coefficient matrix; then identify the type and stability of the critical point.

b) Plot the vector field on a region small enough to distinguish the critical points but large enough to judge the possible solution behaviors away from the critical points.

c) Use several initial data points (x0, y0) in the first quadrant to draw a phase portrait for the system. Identify the direction of increasing t on the trajectories you obtain. Use the information from parts (a) and (b) to choose a representative sample of initial conditions. Then combine the vector field and phase portrait on a single graph.

d) Explain from your phase portrait how the populations vary over time for initial data close to the unique critical point inside the first quadrant. What happens for initial data far from this critical point?

e) Suppose the initial state of the population is given by x(0) = 1, y(0) = 1 Find the state of the population at t = 1, 2, 3, 4, 5. f) Estimate to two decimal places the period of the solution curve that starts at (1, 1)

Explain what the following notions mean for hash functions and compare their strength. Please don’t copy the definitions from the class slides, explain them with your own words.

• Collision Resistant

• Pre-image Resistant

• Second Pre-image Resistant

During a flu epidemic in a city of 1,000,000, a flu vaccine sells in proportion to both the number of people already inoculated and the number not yet inoculated. If 100 doses have been sold at time t = 0and after 5 weeks 8000 doses have been sold, find a formula for the total number of doses sold within t weeks.

y=

Abox containing pennies, nickels, and dimes has 13 coins with a total value of 83 cents. How many coins of each type are in the box? Is the economy productive?

Using (gauss elminations)

5. (a) What are the monthly payments on a 20-year, $200,000 loan having 6% annual interest, compounded monthly? Interest is computed based on the balance at the beginning of each month, and payments are made at the end of eachmonth.

(b) How much must be deposited each month to achieve a balance of $200,000 at the end of 25 years, at 6% annual interest compounded monthly? Deposits are made at the beginning of each month, and interest is paid on the balance at the end of each month.

how many integers from 0 through 999,999 contain the digit 4 exactly twice?

how many integers from 1 through 1000000 contain the digits 6 at least once

Problem 3: Find a second solution by reduction of order - nonhomogeneous

The given function y1(x) is a solution of the associated homogeneous equation. Use the reduction of order method to find a solution y(x) = u(x)y1(x) of the nonhomogeneous equation.

1. x^2y'' + xy'−4y = x^3, y1 = x^2

2. 2x^2*y'' + 3xy'−y = 1/x ,    y1 = x^(1/2)

I am confused on the Gram-Schmidt Orthogonalization Process and how it works? The examples do not make sense to me. Thank You.

Discuss the steps you would need to take in order to make your practicum classroom a STEM learning environment. Support your steps with research.

Explain the questions with detail and how to solve it:

1) Ahmed can paint the office in 5 hours. When Jamel helps him it takes them 4 hours working together. How long will it take Jamel if he works alone?

2) If a projectile is fired straight upward form the ground with an initial speed of 96 feet per second, then its height h in feet after t seconds is given by the function h(t)= -16t^2+96t
Find the maximum height of the projectile.

3) he model A= 1.6 e^0.039t gives the minimum wage, A, in year t, and where t= 0 represents the year 1970.
a)Use model to predict the minimum wage in the year 2018.
b) Use the model to find the year in which the minimum wage reach $15 per hour.