Find first four non zero terms in a power series expansion about x=0 for a general solution to the given differential equation.

(x^2 +21)y''+y=0

What is different about SL(2,Z) and GL(2,Z) with regard to its action on the Farey treeing particular distinguished edge. (Geometric group theory.)

John’s Construction has three projects under way. Each project requires a regular supply of gravel, which can be obtained from three quarries. Shipping costs differ from location to location, and are summarized in the table.

Formulate a transportation model (but do not attempt to solve it) which could be used to determine the amount of gravel to be shipped from each quarry to the various job sites.

Consider the following model: maximize 40x1 +50x2 subject to: x1 +2x2 ≤ 40 4x1 +3x2 ≤ 120 x1, x2 ≥ 0 The optimal solution, determined by the two binding constraints, is x1 = 24, x2 = 8, OFV∗ = 1,360. Now consider a more general objective function, c1x1 + c2x2. Perform a sensitivity analysis to determine when the current solution remains optimal in the following cases: (i) both c1 and c2 may vary; (ii) c2 = 50, c1 may vary; (iii) c1 = 40, c2 may vary. Suppose the RHS of the second constraint increases by an amount ∆b. (It is now 120 + ∆b.) Solve the two equations for x1 and x2 in terms of ∆b, and hence determine its shadow price.

1.Determine which amounts of postage can be formed using just 3-cent and 10-cent stamps.

2.Prove your answer to (a) using the principle of mathematical induction. Be sure to state explicitly your inductive hypothesis in the inductive step.

3.Prove your answer to (a) using strong induction. How does the inductive hypothesis in this proof differ from that in the inductive hypothesis for a proof using mathematical induction?

q = a(b+c(de+f)), t = !(q+a), m = aq, where the inputs are a, b, c, d, e, f and q; and outputs are t and m.

Please provide the truth table.

Please write the Sum of Products for the output t.

Implement the Sum of Products of t with AND2, OR2 and NOT gates.

Convert the previous part by using ONLY NAND2 gates.

Provide MIPS code for the last part.

Give examples of practical and theoretical math from 1700s,1800s,1900s. What was the interaction between the theoretical and the applied. What extent practical considerations at times get ahead of theoretical.

S(x) is a cubic spline for the function f(x) = sin(pi x/2) + cos(pi x/2) at the nodes x_{0 ⁼} 0 , x₁ = 1 , x₂ = 2

and satisfies the clamped boundary conditions. Determine the coefficient of x³ in S(x) on [0,1] ans. pi/2 -3/2

Consider rolling two 6-sided dice. What is the probability that

1. at least two of the rolls have a sum that exceeds 6?
2. at least 7 of the rolls have a sum that is even?
3. exactly three rolls have a sum that equals 5?

Give an example of a continuous function that is not uniformly continuous. Be specific about the domain of the function.

prove that if f is a univalent function in D then w=f(z) is conformal mapping in every point in D

Please solve the following equation by using the frobenius method.

xy′ = (x + 1)y

My apologies, the original image did not upload. Thank you!

Use the graph to find the limit L (if it exists). If the limit does not exist, explain why. (If an answer does not exist, enter DNE.)

h(x) = -x/2 + x²

(a)

lim x→2 h(x)

L =

(Select One)

The limit does not exist at x = 2 because the function is not continuous at any x value.

The limit does not exist at x = 2 because the function approaches different values from the left and right side of 2.    

The limit does not exist at x = 2 because the function value is undefined at x = 2.

The limit does not exist at x = 2 because the function does not approach f(2) as x approaches 2.The limit exists at x = 2.

(b)

lim x→1 h(x)

L =

(Select One)

The limit does not exist at x = 1 because the function does not approach f(1) as x approaches 1.

The limit does not exist at x = 1 because the function approaches different values from the left and right side of 1.    

The limit does not exist at x = 1 because the function is not continuous at any x value.

The limit does not exist at x = 1 because the function value is undefined at x = 1.The limit exists at x = 1.

Find the solution of the given initial value problem:

3y′′′+27y′−810y=0

y(0)=11, y′(0)=39, y′′(0)=−261