What is a Bessel Differential Equation? Classify Bessel Equation with respect to its order, linearity and homogeneity. How many independent solutions of Bessel Differential equation are needed to construct general solution?

One of the luxury car manufacturers usually wants to make a limited number of automobiles in a given year and the company’s aim is to expand their profit margins in each car sold. After doing the analysis on the sales of their cars, the company determined the below supply and demand functions of the cars sold:

Supply function: P = 1000 + Q^2

Demand function: P = 21000 – Q^2

a) Calculate the producer, consumer and total surplus at the market equilibrium.

b) If the manufacturer restricts the number of cars manufactured and sold to 800 at the demand price of $15,000, how much is the total producer surplus?

c) A manufacturer wants to maximize the producer surplus and hence determine the quantity (give the whole number) at which this can be possible. Also determine the producer surplus at this quantity.

(a) A bank’s safe has 5 dials each of which can be set to a number from 0 to 36 (so each dial has 37 possible settings).

i. How many different ways can the 5 dials be set?

ii. How many settings are there if no two dials may be set to the same number?

iii. How many possible settings are there if all the numbers must be even (count 0 as even), and duplicate numbers are permitted?

iv. Re-do the previous question, but this time with no duplicates allowed.

(b) Suppose the bank gets a new safe also with 5 dials, but each dial is numbered as follows: Dial 1 goes from 0 to 10 Dial 2 goes from 0 to 12 Dial 3 goes from 0 to 20 Dial 4 goes from 0 to 25 Dial 5 goes from 0 to 35

i. How many different ways can the 5 dials be set?

ii. How many settings are there if no two dials may be set to the same number?

iii. How many possible settings are there if all the numbers must be even (count 0 as even), and duplicate numbers are permitted?

iv. Re-do the previous question, but this time with no duplicates allowed.

Find the general solution of the given differential equation, and use it to determine how solutions behave as t→∞.

9y′+y=5t2

Find a power series solution of this ODE:

y''+x^2y'=0 ; y(0)=1 y'(0)= 2

Prove verbally without referencing a Venn diagram, can use a Vennn diagram to visually see it but cannot be a reference in proof.

a) Let A be a set. Prove that A⊆ A.

b) Let A, B, and U be sets so that A ⊆ U and B ⊆ U. Prove that A ⊆ B if and only if (U \ B) ⊆ (U \ A)

c) Let A, B be sets. Prove that A \ (A ∩ B) = A \ B

d) Let A, B be sets. Prove that A \ (B \ A) = A

1)

A full rectangular water tank has a base of 10 ft x 15 ft and is 20 ft high. What is the pressure at the bottom of the tank in a) PSI b) Pa

2)

A cargo ship has a volume of 10,000 cubic meters. The total mass of the ship and cargo is 8 million kg. Will it sink or float at the surface of seawater having a density 1025 kg/m3? Why?

what would be some practical research scenarios in the field of Social Work where this type of statistical test would be appropriate.

Prove that if f, g are integrable, then the function (f(x) + cos(x)g(x))2is integrable.

From a shipment of 75 transistors, 5 of which are defective, a sample of 6 transistors is selected at random.

(a) In how many different ways can the sample be selected?

(b) How many samples contain exactly 3 defective transistors?

(c) How many samples do not contain any defective transistors?

I have no clue how to do the math here and get the values asked? Please help by working out the equation.  

Two species of algae (called Blue and Green) compete. Their dynamics can be described using the Lotka-Volterra competition equations:

dN_B/dt = r_BN_B(K_B-N_B- a_BGN_G)/K_B

dN_G/dt = r_GN_G(K_G-N_G- a_GBN_B)/K_G                                                                     

The following information is known:

dN_B/dt = 0 when N_B = 100 and N_G = 0

dN_B/dt = 0 when N_B = 50 and N_G = 25

dN_G/dt = 0 when N_B = 50 and N_G = 25

dN_G/dt = 0 when N_B = 0 and N_G = 225

a. (6 pts.) Using this information, determine the values of:

K_B =

K_G =

a_BG =

a_GB =

Consider the differential equation:

y'(x)+3xy+y^2=0.     y(1)=0.    h=0.1

Solve the differential equation to determine y(1.3) using:

a. Euler Method
b. Second order Taylor series method
c. Second order Runge Kutta method
d. Fourth order Runge-Kutta method
e. Heun’s predictor corrector method
f. Midpoint method

Calculate the Euler method approximation to the solution of the initial value problem at the given x-values. Compare your results to the exact solution at these x-values.

y' = y+y^2; y(1) = -1, x = 1.2, 1.4, 1.6, 1.8

Determine the reasonable form of the particular solution for each non homogeneous differential equation. Do not solve it.

a) y''-y'-2y= e^-x+xcos2x+e^xsin2x.

b) D^2[y] +4y =1+x^2+xsin2x.