8) Solve x^2y’’ – 2y = ln(x)

2)   Solve the system of equations below
          dx/dt – 3x – 6y = t^2
          dx/dt + dy/dt – 3y = e^t

Use the method of variation of parameters to find the complete solution of the differential equation d²y/ dx² + 4 dy /dx + 4y = e ^−2x ln(x), x > 0.

Q: Asking for assistance in understanding and solving this example on Modern Algebra II with the steps of the solution to better understand, thanks.

**Please give the step by steps with details to completely see how the solution came about.

1) Determine all elements in an integral domain that are their own inverses under multiplication.

2) Let F be a finite field with n elements. Prove that x^(n-1 )= 1 for all nonzero x in F. Hint: Use the fact that the nonzero elements of F form a group under the multiplication operation.

Please show picture over a rectangle.

a) Let f(x,y) = 2sin(πx)−3cos(πy). Calculate Uf(P) and Lf(P) over the partition P = {1,1.5,2}×{2,2.5,3}.

b) Explain what techniques would be necessary to calculate Uf(P) and Lf(P) for f(x,y) = 2sin(x)−3cos(y) over the partition P = {1,1.5,3}×{2,2.5,3}.

what is the tenth letter of the standard english alphabet?

Let f : Z × Z → Z be defined by f(n, m) = n − m

a. Is this function one to one? Prove your result.

b. Is this function onto Z? Prove your result

Give a clear simple argument that each of the following sets is uncountable.

a. R × Q

b. N ∪ P(N)

c. P(R)

d. (0, π) ∪ {4, 5, 6, 7}

Solve for x,y,z using the inverse if possible.

x+2y+5z=2

2x+3y+8z=3

-x+y+2z=3

describe a project list two task that may be performed in parallel and two tasks that need to be performed sequeooontially

Containers for liquids have interior volumes that are larger than their rated capacity. This is necessary to allow for expansion of the liquid contents as the temperature changes. You have a five gallon steel fuel can manufactured to have an interior volume of 5.22 gallons (19.76 L). The coefficient of thermal expansion of steel is small enough that this volume will not change significantly with temperature in this problem.

One cold spring morning, you pump 5.00 gallons (18.93 L) of gasoline into your fuel can, leaving 0.22 gallons (0.83 L) unfilled. The gasoline comes from an underground tank at 54.9°F (12.7°C, 285.9 K), but the outside temperature is 31°F (-0.6°C, 272.6 K). You then screw on the cap tightly and leave the can outside. The weather forecast calls for the temperature to stay constant in the morning morning, followed by a significant warming trend in the afternoon, when the temperature will rise to a high of 75°F (23.9°C, 297.0 K). The atmospheric pressure will stay constant at 973 mbar (97.3 kPa) all day.

1. Gasoline has a coefficient of thermal expansion of 9.50 x 10^-4/°C. What will the unfilled volume in your fuel can be after the gasoline cools in the morning?

2. What will the absolute pressure of the air in the sealed fuel can be then?

3. What will the unfilled volume in your fuel can be when the gasoline and air in the fuel can warm to the afternoon high temperature?

4. What will the absolute pressure of the air in the sealed fuel can be then, assuming no air can escape?

5. Your fuel can actually has a safety gasket which will allow air to leave the can if the pressure difference to the outside (called the “gauge pressure”) exceeds 5.0 psi (34 kPa). Will the safety gasket vent air from the fuel can in the afternoon?

The concept of “market equilibrium” is defined as when the quantity of a commodity demanded is equal to the quantity supplied. Assume that the demand function and the supply function are both linear. How can good advertising affect market equilibrium? How can bad advertising affect market equilibrium?

solve y" + 4y = 5 tan(2x)

Use power series to find two linearly independent solutions centered at the point x=0

1) y'' + 2y' - 2y = 0

2) 2x²y'' + x(x-1)y' - 2y = 0

please show work, thank you!

Solve the given Boundary Value Problem. Apply the method undetermined coefficients when you solve for the particular solution.

y′′+2y′+y=(e^-x)(cosx−sinx)

y(0)=0,y(π)=e^π