Consider the IVP t²y''−(t² + 2)y' + (t + 2)y = t³ with y(1) = 0 and y'(1) = 0.

• One function in the fundamental set of solutions is y₁(t) = t. Find the second function y₂(t) by setting y₂(t) = w(t)y₁(t) for w(t) to be determined.

• Find the solution of the IVP

Consider the IVP t²y''−(t² + 2)y' + (t + 2)y = t³ with y(1) = 0 and y'(1) = 0.

• One function in the fundamental set of solutions is y₁(t) = t. Find the second function y₂(t) by setting y₂(t) = w(t)y₁(t) for w(t) to be determined.

• Find the solution of the IVP

A company manufactures Products A, B, and C. Each product is processed in three departments: I, II, and III. The total available labor-hours per week for Departments I, II, and III are 900, 1080, and 840, respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows. (For example, to make 1 unit of product A requires 2 hours of work from Dept. I, 3 hours of work from Dept. II, and 2 hours of work from Dept. III.)

How many units of each product should the company produce in order to maximize its profit?

What is the largest profit the company can realize?
$  

Are there any resources left over? (If so, enter the amount remaining. If not, enter 0.)

15. Let r be a positive real number. The equation for a circle of radius r whose center is the origin is (x^2)+(y^2)= r^2 .

(a) Use implicit differentiation to determine dy/dx .

(b) Let (a,b) be a point on the circle with a does not equal 0 and b does not equal 0. Determine the slope of the line tangent to the circle at the point (a,b).

(c) Prove that the radius of the circle to the point (a,b) is perpendicular to the line tangent to the circle at the point (a,b).

Hint: Two lines (neither of which is horizontal) are perpendicular if and only if the products of their slopes is equal to

The integers satisfy a property known as mathematical induction. This is a familiar topic in high school textbooks.

(a) The First Principle of Mathematical Induction is stated as follows. Suppose S is a subset of N with the following properties: (i) The number 1 is in S. (ii) If n is in S, then n + 1 is in S. Using well-ordering, prove S = N.

(b) The Second Principle of Mathematical Induction is stated as follows. Suppose S is a subset of N with the following properties: (i) The number 1 is in S. (ii) If n is in S and if every natural number k, where k ≤ n, is in S, then n + 1 is in S. Using well-ordering, prove S = N.

Show, using mathematical induction, that 2k−1 ≤ k! for all natural numbers k, by doing parts (a) and (b).

(a) Let S be the set of natural numbers for which the inequality is true. Show that 1 is in S.

(b) Now conclude, using the induction hypothesis and the theorems about inequalities, that 2 n = 2 · 2 n−1 ≤ 2 · n! ≤ (n + 1) · n! = (n + 1)! That is, if n is in S, then n + 1 is in S.

Define the following order on the set Z × Z: (a, b) < (c, d) if either a < c or a = c and b < d. This is referred to as the dictionary order on Z × Z.

(a) Show that there are infinitely many elements (x, y) in Z × Z satisfying the inequalities (0, 0) < (x, y) < (1, 1).

(b) Show that Axioms O1–O3 ( Trichotomy, Transitivity, Addition for inequalities) are satisfied for this ordering.

(c) Give an example that shows that Axiom O4 (Multiplication for inequalities) is not satisfied for this ordering.

(d) Is the well-ordering axiom satisfied for Z × Z with the dictionary order?

Let G be an abelian group and n a fixed positive integer. Prove that the following sets are subgroups of G.

(a) P(G, n) = {gⁿ | g ∈ G}.

(b) T(G, n) = {g ∈ G | gⁿ = 1}.

(c) Compute P(G, 2) and T(G, 2) if G = C₈ × C₂.

(d) Prove that T(G, 2) is not a subgroup of G = D_n for n ≥ 3 (i.e the statement above is false when G is not abelian).

1) Find the solution of the given initial value problem and describe the behavior of the solution as t → +∞

y" + 4y' + 3y = 0, y(0) = 2, y'(0) = −1.

2) Find a differential equation whose general solution is Y=c₁e^2t + c₂e^-3t

3) Determine the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution. Do not attempt to find the solution t(t − 4)y" + 3ty' + 4y = 2 = 0, y(3) = 0, y'(3) = −1.

4) Consider the ODE: y" + y' − 2y = 0. Find the fundamental set of solutions y1, y2 satisfying y1(0) = 1, y'1 (0) = 0, y2(0) = 0, y'2 (0) = 1.

Determine the number of permutations of {1,2,3,...,n-1,n} where n is any positive integer and no even integer is in its natural position.

A certain college graduate borrows $8,937 to buy a car. The lender charges interest at an annual rate of 16% . Assuming that interest is compounded continuously and that the borrower makes payments continuously at a constant annual rate, determine the payment rate that is required to pay off the loan in 6 years. Also determine how much interest is paid during the 6 -year period. When calculating the interest use the non-rounded value of the payment rate, then round all answers to two decimal places.

Payment rate = $_____ per year

Interest paid = $ ______ per year

This question is for a differential equations course.

A batch of 50 different automatic typewriters contains exactly 10 defective machines. What is the probability of finding:

(c) The first defective machine to be the k-th machine taken apart for inspection in a random sequence of machines?
(d) The last defective machine to be the k-th machine taken apart?

Find the maximum of f(?₁,?₂)=4x₁+ 2?₂ + ?₁² − 2?₁⁴ + 2?₁?₂ − 3?₂² using the steepest ascent method with initial guess ?₁=0 and ?₂=0.
(a) Find the gradient vector and Hessian matrix for the function f(?₁,?₂).
(b) Perform two iterations of the steepest ascent method.

Consider a 5x5 chessboard. Prove that no matter how the 25 cells are colored in red and blue (each cell is either red or blue), there exist 4 cells of the same color whose centers determine a rectangle with sides parallel to the sides of the board. Is the statement true for a 4x4 chessboard? What about 4x6 chessboard?

Use the Frobenius method to solve:xy"+xy^'+3y=0. Find index r and recurrence formulas. Compute the first 5 terms using the recurrence formula for each solution and index r.

The other answer on this website has the poorest handwriting, cant tell between r y x or n. Please make sure it is legible