. Let U be a non-empty set. For A and B subsets of U, define the relation A R B if an only if A is a proper subest of B. a. Is R reflexive? Prove or explain why not. b. Is R symmetric? Prove or explain why not c. Is R transitive? Prove or explain why not. d. Is R antisymmetric? Prove or explain why not. e. Is R an equivalence relation? Prove or explain why no

Solve these first-order Differential Equations using an integrating factor.

1. dy/dx+2xy=0

2. dy/dx-y=5

3. dy/dx+y=x

4. (x)dy/dx+(x+1)y=3/x

5. (x^2)dy/dx=e^x-2xy

Is the following argument p& ∼ p,(p ∨ q) ≡ (s ≡ (t ∨ p)) ∴ r valid? If not provide a counterexample.

A new type of fueling truck is under consideration for an airport on a resort island. The company has been testing one alternative and feel it has the potential to significantly reduce fueling time while airplanes are at the airport’s one gate. With current equipment refueling requires 20 minutes. They estimate that the first fueling operation with this equipment will require 30 minutes. They hope, by spending the money for the new equipment, that within two weeks they will achieve a refueling time of 15 minutes – a 25% improvement on the current time. Assume 6 planes per day are schedule to arrive/depart from the island (7 days/week). All flights to and from the island are scheduled to arrive and depart between 6:00 am and 9:00 am.

a. Determine the learning rate required to achieve their objective.

b. If their learning rate is actually 92%, how many days will it take to get the refueling time below the 20 minutes required with the old system?

c. After 10 weeks with a learning rate of 92%, what would be the expected time to refuel planes with the new system?

d. What impact (if any) will this change have on the demand for fueling systems and the way they are scheduled?

e. What impact might this have on the scheduling of flights to/from this destination?

f. Assuming there’s demand for up to six additional flights in the scheduling window, what is the business case for purchasing this equipment?

solve the initial values:

if Y(3)-4Y"+20Y'=51e^3x

Y"(0)=41, Y'(0)= 11. Y(0)= 7 > solution is Y(x)= e^3x+2 e^2x sin(4x)+6

so, what is the solution for:

Y(3)-8Y"+17Y'=12e^3x

Y"(0)=26, Y'(0)= 7. Y(0)= 6

Y(x)=???

Consider the initial-value problem y' = 2x − 3y + 1, y(1) = 7. The analytic solution is y(x) = 1/9 + 2/3 x + (56/9) e^(−3(x − 1)).

(a) Find a formula involving c and h for the local truncation error in the nth step if Euler's method is used.

(b) Find a bound for the local truncation error in each step if h = 0.1 is used to approximate y(1.5). (Proceed as in this example.)

(c) Approximate y(1.5) using h = 0.1 and h = 0.05 with Euler's method. (Round your answers to four decimal places.)

h = 0.1          y(1.5) ≈______

h = 0.05         y(1.5) ≈______

(d) Calculate the errors in part (c) and verify that the global truncation error of Euler's method is O(h). (Round your answers to four decimal places.) Since y(1.5) =______, the error for h = 0.1 is E_0.1 = ______, while the error for h = 0.05 is E_0.05 = ______. With global truncation error O(h) we expect E_0.1/E_0.05 ≈ 2. We actually have E_0.1/E_0.05 = _______ (rounded to two decimal places).

1. Find the first four terms in each portion of the series solution around x₀ = 0 for the following differential equation. x² y^// - 5x y^/ + 6y = 0

Differential Geometry

Open & Closed Sets, Continuity

Prove f(t)=(x(t),y(t)) is continuous iff x(t) and y(t) are continuous

(1) Show that the set { 1 m + 1 n : m, n ∈ N} is countable.

(2) Show that the set {a + b √ 2 : a, b ∈ Q} is countable.

(3) Show that the intersection of two countable sets is countable.

(4) Show that the set of all irrational numbers is uncountable.

(5) Let C = {0, 1, 2, . . . , 9}. Show that the set C ×C × · · · is uncountable. [Hint: Imitate the proof we had for E × E × · · ·, where E = {0, 1}.]

Partial Differential Equations

(a) Find the general solution to the given partial differential equation and (b) use it to find the solution satisfying the given initial data.

Exercise 1. 2∂u ∂x − ∂u ∂y = (x + y)u

u(x, x) = e −x 2

Exercise 2. ∂u ∂x = −(2x + y) ∂u ∂y

u(0, y) = 1 + y 2

Exercise 3. y ∂u ∂x + x ∂u ∂y = 0

u(x, 0) = x 4

Exercise 4. ∂u ∂x + 2y ∂u ∂y = e −x − u

u(0, y) = arctan y

Exercise 5. ∂u ∂x+v ∂u ∂y = −ru

(here r and v 6= 0 are constants) u(x, 0) = sin x x

Find the eigenvalues and eigenfunctions for the following boundary value problem.

y"+6y'-(L-8)=0, y(0)=0, y(2)=0 L == Lambda

By sepration of variables Solve

a- 9U_yy=6U_xU_y

b- 4U_xx=6U_xy

Consider the ODE y"+ 4 y'+ 4 y = 5 e^(− 2 x ). (

a) Verify that y 1 ( x) = e − 2 x and y 2 ( x) = xe − 2 x satisfy the corresponding homogeneous equation.

(b) Use the Superposition Principle, with appropriate coefficients, to state the general solution y h ( x ) of the corresponding homogeneous equation.

(c) Verify that y p ( x) = 52 x 2 e − 2 x is a particular solution to the given nonhomogeneous ODE.

(d) Use the Nonhomogeneous Principle to write the general solution y ( x ) to the nonhomogeneous ODE.

(e) Solve the IVP consisting of the nonhomogeneous ODE and the initial conditions y(0) = 1 , y 0 (0) = − 1 .

Describe a problem that you are currently faced with at work or in your personal life that could be solved by using an optimization model. Describe what the problem is, why optimization modeling could help you and how you would approach solving the problem with an optimization model.