2.) By Theorem 3.23 of the text, the linear diophantine equation of the form ax + by = c has no integral solutions if c is not divisible by (a, b), the greatest common divisor of a and b. On the other hand if (a, b) divides c, then we can use the Extended Euclidean Algorithm to find integers s, t such that sa + tb = (a, b); multiplying through by the correct factor gives an integral solution x, y. Write a Mathematica procedure that solves any linear diophantine equation of the form ax + by = c, whenever it is solvable. You should invoke your Extended Euclidean Algorithm.

x*y' '+y'-x*y=0

1> use Frobenious series solution to solve the ODE.

Solve the linear programming problem by the method of corners.

Use the method of Undetermined Coefficients to find the solution of the initial value value problem:

y'' + 8y' + 20y = 9cos(2t) - 18e^-4t, y(0) = 5. y'(0) = 0

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}