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.

Let a and b be integers and consider (a) and (b) the ideals they generate. Describe the intersection of (a) and (b), the product of (a) and (b), the sum of (a) and (b) and the Ideal quotient (aZ:bZ).

1. Prove the Heine-Borel Theorem (Theorem 3.35).
2. Suppose f: X → Y maps from the metric space X to the metric space Y, and x ∈ X.
Prove that f is continuous at x if and only if, for any sequence {xn} in X that converges to x, f(xn) → f(x).

Solve SVM for a data set with 3 data instances in 2 dimensions: (1,1,+), (-1,1,-),(0,-1,-). Here the first 2 number are the 2-dimension coordinates. ‘ +’ in 3rd place is positive class. And ‘-‘ in 3rd place is negative class . Your task is to compute alpha’s, w, b. Then, Solve SVM when data are non-separable, using k=2 when minimizing the violations of the mis-classification, i.e., on those slack variables.

For p = 5 , find a number x such that x^2 is congurent to -1 mod p. here x is denoted by sqrt(-1). Determine if sqrt(-1) exists mod p for p = 7, 11, 13 , 17, 23, 29.

How many ways are there to choose three different numbers each between one and a
hundred so that their sum is even? Explain

Prove the following:

Let V and W be vector spaces of equal (finite) dimension, and let T: V → W be linear. Then the following are equivalent.

(a) T is one-to-one.

(b) T is onto.

(c) Rank(T) = dim(V).

2.31. Show that for each of the following values of a and b, there exists x, y in Z satisfying ax + by = 11. (i) a = 11, b = 0, (ii) a = 22, b = 11, (iii) a = 33, b = 22, (iv) a = 451, b = 33, (v) a = 484, b = 451.

2.39. Prove that gcd(ad, bd) = |d|gcd(a, b).

2.44. Does the Diophantine equation 12x + 33y = 1 have an integer solution? If so, can you list all integer solutions?

2.47. For nonzero integers a, b, c, gcd(a, b, , c) denotes the largest integer that divides all of them. Show that gcd(a, b, c) = gcd(a, gcd(b, c)).

Coding: Use MATLAB to figure out the following problem, if you do not know how to use MATLAB then please do not answer. Coding is required for the exercise.

For f(x) = arctan(x), find its zeros by implimenting Newtons method and the Secant method in Matlab. (Hint: Use Newtons method to calculate x1 for Secant method)

Comment all code please since I would like to learn how to do this correctly in MATLAB. Thank you.

Let V be the vector space of all functions f : R → R. Consider the subspace W spanned by {sin(x), cos(x), e^x , e^−x}. The function T : W → W given by taking the derivative is a linear transformation

a) B = {sin(x), cos(x), e^x , e^−x} is a basis for W. Find the matrix for T relative to B.

b)Find all the eigenvalues of the matrix you found in the previous part and describe their eigenvectors. (One of the factors of the characteristic polynomial will be λ 2+1. Just ignore this since it has imaginary roots)

d) Use your answer to the previous part to find all the eigenvalues of T and describe their eigenvectors. Check that the functions you found are indeed eigenvectors of T.

Consider the lattice of real numbers in the interval [0,1] with the relation ≤. Does this lattice have any atoms?

Karon went to the post office and spent $31.57 on 84 stamps. She bought 27¢ stamps for all the postcards, 42¢ stamps for each letter addressed to friends in the U.S., and 94¢ stamps for each letter to England. Karon bought 7 times as many postcard stamps as those for the letters to England. How many postcard stamps did Karon buy? (Hint: Use matrices.)

You need to complete the following SWOT analysis:

Creative Computing sells a tablet computer called the Protab. The $780 sales price of a Protab Package includes the following:

• One Protab computer.
• A 6-month limited warranty. This warranty guarantees that Creative will cover any costs that arise due to repairs or replacements associated with defective products for up to six months.
• A coupon to purchase a Creative Probook e-book reader for $420, a price that represents a 30% discount from the regular Probook price of $600. It is expected that 20% of the discount coupons will be utilized.
• A coupon to purchase a one-year extended warranty for $45. Customers can buy the extended warranty for $70 at other times if they do not use the $45 coupon. Creative estimates that 35% of customers will purchase an extended warranty.
• Creative does not sell the Protab without the limited warranty, option to purchase a Probook, and the option to purchase an extended warranty, but estimates that if it did so, a Protab alone would sell for $760.

Required:
1. & 2. Indicate below whether each item is a separate performance obligation and allocate the transaction price of 100,000 Protab Packages to the separate performance obligations in the contract.
3. Prepare a journal entry to record sales of 100,000 Protab Packages (ignore any sales of extended warranties).

Complete this question by entering your answers in the tabs below.

• Req 1 and 2
• Req 3

Indicate below whether each item is a separate performance obligation and allocate the transaction price of 100,000 Protab Packages to the separate performance obligations in the contract.

• Req 1 and 2
• Req 3

y₁(t) = (1+t)² is a solution to y'' + p(t)y' + q(t)y = 0. Find a second solution that is linearly indepentent of y₁(t).