Find the solution of the given initial value problem.

y⁽⁴⁾ + 2y''' + y'' + 8y' − 12y = 12 sin t − e^−t;    y(0) = 7,    y'(0) = 0,    y''(0) = −1,    y'''(0) = 2

Find the green's function and fine the solution

y"+4y=x

y(0)=0, y'(1)=0

find the eigenvalue and eigenfuction for the Sturm-lioville system

y"+ky=0

y(0)=y(2pi)

y'(0)=y'(2pi)

Jessica bought a portfolio of five bonds for $50,000. Each of the bonds is a par valued 10,000 6% bond with annual coupons. These five bonds mature at time t=16 through t=20. Find Jessica’s APY.

a) Prove that if X is Hausdorff, then X is T₁

b) Give an example of a space that is T₁ , but not Hausdorff. Prove that the space you give is T₁ and prove it is not Hausdorff.

For each of the following relations, determine if f is
• a function,
• surjective, or
• injective.
Conclude by stating if the relation represents a bijective function.
For each point, state your reasoning in proper sentences.
a) f = {(a, b) ∈ N
2 × N | a ∈ N
2
, a = (a1, a2), b, a1, a2 ∈ N, b = a1a2}
b) f = {(x, y) ∈ S
2
| y = x
2}, where S = {x ∈ R | x ≥ 0}

Solve the following differential equations.

a.) (2xy^2 +2x)dx−(4x^2 +1)dy=0

b.) (3ye^(3xy) +4xy)dx+(3xe^(3xy) +2x^2)dy=0

Find two linearly independent power series solutions for the following differential equation. Write the first four terms for each.

y′′ − xy = 0

1. An 8 pound weight stretches a spring 2 feet. The surrounding medium offers a damping force that is numerically equal to 2 times the instantaneous velocity. It is then released from rest from a point 3 feet below the equilibrium point.
a. Determine the equation of motion.

b. Is the system underdamped, overdamped, or critically damped?

Q11: Use the Lagrange interpolating polynomial of degree three or less and four-digit chopping arithmetic to approximate cos 0.750 using the following values. Find an error bound for the approximation.
cos 0.698 = 0.7661 ,cos 0.733 = 0.7432 cos 0.768 = 0.7193 cos 0.803 = 0.6946.

Use the Laplace transform to solve the given initial value problem.

y⁽⁴⁾ − 4y''' + 6y'' − 4y' + y = 0;

y(0) = 1,

y'(0) = 0,

y''(0) = 0,

y'''(0) = 1

For this discussion, you will reflect on the many applications and uses of statistics. Develop a main response in which you address the following:

• Identify two (or more) possible applications of statistics.
• Explain how statistics are beneficial and useful in these situations.
• Discuss what additional information statistics can provide.

d^2y/dx^2 − dy/dx − 3/4 y = 0,

y(0) = 1, dy/dx(0) = 0,

Convert the initial value problem into a set of two coupled first-order initial value problems

and find the exact solution to the differential equatiion

Use mathematical induction to prove that for each integer n ≥ 4, 5ⁿ ≥ 2²ⁿ⁺¹ + 100.

solve the following using Siri Solutions. Verify your solution using usual methods (if possible).

1. y" - 2y' + y = 0

2. y" - 2xy' + y = 0