Find the general solution using REDUCTION OF ORDER. and the Ansatz of the form y=uy1 = y=ue^-x

(2x+1)y'' -2y' - (2x+3)y = (2x+1)² ; y1 = e^-x

^{Thank you in advance}

Provide an example:

1) A sequence with infinitely many terms equal to 1 and infinitely many terms that are not equal to 1 that is convergent.

2) A sequence that converges to 1 and has exactly one term equal to 1.

3) A sequence that converges to 1, but all of its terms are irrational numbers.

Diane has decided to play the following game of chance. She places a $1 bet on each repeated play of the game in which the probability of her winning $1 is .6. She has further decided to continue playing the game until she has either accumulated a total of $3 or has lost all her money. What is the probability that Diane will eventually leave the game a winner if she started with a capital of $1? Of $2?

Find the work done in moving a particle once around an ellipse C in the XY- plane if the ellipse has a center at the origin with semi-major axis p and semi-minor axis 2p and if the force field is given by F= (3x - 4y + 2z)i + (4x +2y - 3z^2)j + (2xz - 4y^2+z^3)k . where p=4

Prove that the covariant derivative of an arbitrary tensor is a tensor of which the covariant order exceeds that of the original by one.

Here are five different functional models that might represent the growth of the number of Zika cases, where x represents the week number, and y represents the number of cumulative cases.

1. Linear y = 258.74x

2. Logarithmic y=937.37ln(x)-202.03

3. Quadratic y = 31.357x ^(2 )+ 134.93x − 122

4. Power y = 55.278x ^2.0101

5. Exponential y=47.399e^(0.6737x)

For each function model listed, create a graph for 1 ≤ x ≤ 5, along with the Zika case data from Step 2. Be sure your graphs clearly label the function and the actual data.

Step 2 that the question is referring to is

Below is data of the weekly number of suspected Zika cases in French Polynesia in 2013.Plot the data on the graph below.

Consider the IVPs: 
        (A)  y'+2y = 1,   0<t<1 ,  y(0)=2.
        (B)  y' = y(1-y), 0<t<1 ,  y(0)=1/2.

1. For each one, do the following:
  a. Find the exact solution y(t) and evaluate it at t=1.
  b. Apply Euler's method with  Δt=1/4  to find Y4 ≈ y(1).
     Make a table of tn, Yn for n=0,1,2,3,4.
  c. Find the error at t=1.

2. Euler's method is obtained by approximating y'(tn) by a forward finite difference.
   Use the backward difference approximation to y'(tn+1) to derive the 
   Backward Euler Method:   Yn+1 = Yn + Δt f(tn+1, Yn+1) , n=0,1,2,...  

   Note that now the unknown Yn+1 appears inside f(.,.), so this equation needs to be 
   solved for Yn+1 at each time-step!!! whence it is also called Implicit Euler Method.
   For the simple ODEs (A), (B) above, the updating equation can be solved by hand, 
   but in general a root-finder (like Newton-Raphson) is needed.
   This scheme is also 1st order, but it has better stability properties than Explicit Euler.

   For each IVP problem (A), (B), do the following:
  a. Apply the Backward Euler Method with  Δt=1/4  to find Y4 ≈ y(1).
  b. Find the error at t=1 and compare with Explicit Euler.


3. Use the centered difference approximation to y'(tn) to derive the so called 
   Midpoint Method:   Yn+1 = Yn-1 + 2 Δt f(tn, Yn) , n=1,2,... 
 
   Note that this requires both Yn-1 and Yn to produce Yn+1.  It is a 2-step method, 
   hence not self-starting (need Y0 and Y1 before it can be applied), 
   so some single-step method (like Euler) must be used to start it off.
   However, it has the advantage of being a 2nd order method, and explicit.
 
   For each IVP problem (A), (B), do the following:
  a. Apply the Midpoint Method with  Δt=1/4  to find Y4 ≈ y(1).
  b. Find the error at t=1 and compare with Explicit Euler and with Implicit Euler.
     Which method seems to be doing better in this case ?

the table below lists information for seven Cepheid variables where ‘Period’ is the pulsation period of the star, and ?/?⊙ is the luminosity in solar units.

a) Plot the PL relation for these stars using the logarithms of the values in the table (that is, ???(?) vs ???(?/?⊙)) and extract the slope and y-intercept of the relation the data define

Maximize p = 13x + 8y subject to

P = ? (X,Y)= ( ?,? )

Let T denotes the counterclockwise rotation through 60∘, followed by reflection in the line y=x.

(i) Show that T is a linear transformation.

(ii) Write it as a composition of two linear transformations.

(iii) Find the standard matrix of T.

Coefficient of drag (flat plate, NASA) = C(d) = 1.28

Weight = W = 4451 lbs

Gravitation constant = g = 32.2 ft/sec^2

Area = A = 197.5" long x 78.2" wide x (1 ft^2/ 144 in^2)

Vehicle falls flat, wheels 1st, straight down, at constant acceleration with no aerodynamic drag until terminal velocity

Horsepower needed to accelerate is AVERAGE - not peak

100% driveline efficiency

Patio Iron makes wrought iron outdoor dining tables, chairs, and stools. Each table uses 8 feet of a standard width wrought iron, 2 hours of labor for cutting and assembly, and 2 hours of labor for detail and finishing work. Each chair uses 6 feet of the wrought iron, 2 hours of cutting and assembly labor, and 1.5 hours of detail and finishing labor. Each stool uses 1 foot of the wrought iron, 1.5 hours for cutting and assembly, and 0.5 hour for detail and finishing work, and the daily demand for stools is at most 10. Each day Patio Iron has available at most 154 feet of wrought iron, 55 hours for cutting and assembly, and 50 hours for detail and finishing. If the profits are $60 for each dining table, $48 for each chair, and $36 for each stool, how many of each item should be made each day to maximize profit?