Solve cos^2(x)-cos(x)=0 for x,

Question 3: Exact Differential Equation
i. Test to see if the following equation is exact, if exact solve for the general solution
(??/??) +(2? ????+?^3?^?)/(?^2cos?+3?^2?^?)=0

ii. Solve (?+????)??+(?????−2?)??=0

iii. Solve (?^2+?)??+(?^?−?)??=0

iv. (2?−3?)??+(2?−3?)??=0

I am sorry I could not send a photo. Whenever I try from the app it posts another question on the same page. I tried doing them individually and the only thing that happened was that the same question got posted 4 times. I would appreciate your help in solving these. Thank you!

use the definition of the Taylor series to find the first four nonzero terms of the series for f(x) centered at x = a

a) f(x) = xe^x, a = 0

b) f(x) = sin (x), a = π/6

A mass of 1 slug, when attached to a spring, stretches it 2 feet. It is released from a point 1 foot above the equilibrium position with a downward velocity of 2 ft/s.

1) Find the equation of motion if the surrounding medium offers a damping force that is numerically equal to 4 times the instantaneous velocity.

2) Classify the motion as underdamped, overdamped, or critically damped.

we were looking at lives some men mathematicians. But women made very serious contributions to mathematics as well. One of the most important people in the history of mathematics was Emmy Noether. Please research what you can find about her life and contributions, and share a thought.

solve completely(complementary function and particular integral) pdes

1 (1+D^2-D')z = -8siny

2 (D-D'-D'^2)z =132xy^10+504x^2y^6

3 (D^2-4DD' +4D'^2)z = x + y

4 (D^2-2DD'+ D'^2)z = e^2x-2y+x^2

Respond to the following in a minimum of 175 words: How will you graphically represent data in your future career? What types of graphs are typically used in that industry? What types of conclusions are drawn from the graphs?

Suppose A*B*A′ holds(B is between A and A')  and D ∈ Int(∠ABC). Prove that C ∈ Int(∠A′BD).

(a) Prove that C ∈ H(D,line A′B).

(b) Prove that C ∈ H (A′, line←→BD). Use point A.

(c) Deduce that C∈Int (∠A′BD).

14) To purchase studio space, on April 3, 2025 a pianist takes out a 30-year loan of $120,000 at an annual interest rate of 11%. The first monthly payment is to be made on May 3, 2025; the final payment is scheduled to be made April 3, 2055. The monthly payment is $1,142.79 . [Note: If you use the table you’ll get that the monthly payment is $1,142.40, but the table produces an approximation. $1,142.79 is the precise payment.]

a) Calculate the total interest charged over the course of the loan, assuming that nothing but the standard payment is made each month.

b) How much will she owe immediately after making the payment on June 3, 2025? (= the second payment) Round appropriately to the nearest penny.

c) Suppose that as part of that June 3, 2025 payment she pays an extra $300 as an overpayment. Calculate the dollar amount of interest which this saves in the future. Round your answer down to the nearest $10.

For each group in the following list find the order of each element in the group: Z₁₂, U(10) and U(12).

Find the real solutions of the following equation.

x^4 +7x^3+8x^2-7x+15=0

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

The solution set is { }

(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed. Type each answer only once.)

B.

The solution set is

empty set∅.