Suppose y₁ = -6e^3t - 6t² and y₂ = 4e^-2t - 6t² are both solutions of a certain nonhomogeneous equation: y'' + by' + cy = g(t).

a. Is y = 12e^3t - 4e^-2t + 6t² also a solution of the equation?

b. Is y = -6t² also a solution of the equation?

c. Could any constant function y = c also be a solution? If so, find all possible c.

d. What is the general solution of the equation?

e. Determine the coefficients b, c, and g(t) in the equation above.

Write a program that counts how many Fibonacci numbers are divisible by 3 and smaller than 1000. The program prints the resulting number. You may only use while loops.

Use python language

Solve

x^3 -3x^2 +27 = 0(mod1125)

Show that there is only one positive integer k such that no graph contains exactly k spanning trees.

show that the power set of N and R have the same cardinality

According to the Fundamental Theorem of Algebra, every nonconstant polynomial f (x) ∈

C[x] with complex coefficients has a complex root.

1. (a) Prove every nonconstant polynomial with complex coefficients is a product of linear polynomials.

2. (b) Use the result of the previous exercise to prove every nonconstant polynomial with real coefficients is a product of linear and quadratic polynomials with real coefficients.

In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring of polynomials with real coefficients. For the purposes of this exercise, extend the definition of degree by deg(0) = −1. The statement to be proved is: Let f(x),g(x) ∈ R[x][x] be polynomials with g(x) ? 0. Then there exist unique polynomials q(x) and r(x) such that

f (x) = g(x)q(x) + r(x) and deg(r(x)) < deg(g(x)).

Fix general f (x) and g(x).

1. (a) Let S = { f (x) − g(x)s(x) | s(x) ∈ R[x][x]}. Prove that if h1(x) ∈ S and deg(h1(x)) ≥ deg(g(x)), then there is an

   h2(x) ∈ S with deg(h2(x)) < deg(h1(x)).

2. (b) Show: If h1(x), h2(x) ∈ S with deg(h1(x)) = deg(h2(x)), then there is an h3(x) ∈ S with deg(h3(x)) < deg(h1(x)).

3. (c) Prove S has a unique element of minimal degree.

4. (d) Verify the existence of q(x) and r(x).

Explain how and why synthetic division works.

Find the eigenvalues λ_n and eigenfunctions y_n(x) for the given boundary-value problem. (Give your answers in terms of k, making sure that each value of k corresponds to two unique eigenvalues.)

y'' + λy = 0,  y(−π) = 0,  y(π) = 0

λ_{2k − 1} =, k=1,2,3,...

y_{2k − 1}(x) =, k=1,2,3,...

λ_2k =, k=1,2,3,...

y_2k(x) =, k=1,2,3,...

Choose which of the following is the set of limit pints of the half-open interval [1,3).

a. {1,3}

b.[1,3]

c.{3}

d. (1,3)

e. [1,3)

or

f.{1}

One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no offspring. Let P represent the number of female insects in a population and S the number of sterile males introduced each generation. Let r be the per capita rate of production of females by females, provided their chosen mate is not sterile. Then the female population is related to time t by

t =

Suppose an insect population with 10,000 females grows at a rate of

r = 1.8

and 700 sterile males are added. Evaluate the integral to give an equation relating the female population to time. (Note that the resulting equation can't be solved explicitly for P. Remember to use absolute values where appropriate.)

A mass-spring-dashpot system is described by

my′′ + cy′ + ky = F_o cos ωt,

see §3.6 Eq. (17). This second-order differential equation has been used in simulations, such as this

one at the PhET site: https://phet.colorado.edu/en/simulation/legacy/resonance.

For m = 2.53kg, c = 0.502N/(m/s), k = 97.2N/m, F_o = 97.2×0.5N = 48.6N,and ω = 2.6, the equation becomes

2.53y′′ + 0.502y′ + 97.2y = 48.6 cos(ωt) ....................... (1)

(a) Given the initial value

y(0) = 2.20, y′(0) = 0,

solve Eq. (1), Round to three significant figures. Show all your work, and clearly highlight your

conclusion—the combination of complementary function and particular solution.

(b) From your particular solution, which is of the form

y_p =Acosωt+Bsinωt,

calculate the amplitude, which is √(A2 + B2). Clearly highlight your conclusion—the amplitude. You are encouraged to use the PhET simulation to verify your amplitude. Note that the angular frequency ω = 2πf where f is the frequency in the simulation.

If (x,y,z) is a primitive Pythagorean triple, prove that z= 4k+1

What is the general solution to xy''+y'+k^2xy=0