Solve the differential equation

y'''+y''+y'+y=sinx+e^x+e^{-x}

1) find the solution t the non-homogenous DE

y''-16y=3e^5x , y(0)=1 , y'(0)=2

2)find the solution to the DE using cauchy-euler method

x²y''+7xy'+9y=0 , y(1)=2 , y'(1)=3

3)find the solution to the DE using Laplace

y''+8y'+16y=0 , y(0)=-1 , y'(0)=8

1. For each permutationσ of {1,2,··· ,6} write the permutation matrix M(σ) and compute the determinant |m(σ)|, which equals sgn(σ).

(a) The permutation given by 1 → 2, 2 → 4, 3 → 3, 4 → 1, 5 → 6, 6 → 5.

(b)  The permutation given by 1 → 5, 2 → 1, 3 → 2, 4 → 6, 5 → 3, 6 → 4.

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A.) State Product Rule, Quotient Rule, and Chain Rule.

B.) Prove Power Rule

C.) Prove Product rule:

-By definition of derivative

-By implicit differentiation

D.) Prove the Quotient Rule

-By definition of derivative

-By implicit differentiation

-By product rule and chain rule

Please find f(x') and f(x") for all:

1) f(x) = 3(x² -2x)^3/4(7 - 5x²)⁶

2) f(x) = (y - x)⁴(y² - x)³

3) f(x) = x^7/3 + 16x³ + x

4) = f(x) = (x² - x) / (y² - y)

Newton's Law of Cooling is based on the principle that the rate of change of temperature y'(t) of a body in an environment with ambient temperature A is proportional to the difference b/w the temperature y(t) of the body and the ambient temperature A, so that y'=k(A-y) for some constant k.

(1) Sketch the direction field for the equation y'=K(A-y) for k=1/10 and A=70degrees. Then sketch several solution curves based on starting values of y(0) both greater than and less than A. Discuss the implication of your graph in general, and as it relates to the long-term limiting temperature of a cup of hot coffee initially at 190degreeF and a cup of iced coffee initially at 40degreesF in a room with ambient temperature 70degreesF.

Problem 7. Assume that a subset S of polynomials with real coefficients has a property:
If polynomials a(x), b(x) are from S and n(x), m(x) are any two polynomials with real coefficients, then polynomial a(x)n(x) + m(x)n(x) is again in S. Prove that there is a polynomial d(x) from S, such that any other polynomial from S is a multiple of d(x).

a) Prove by induction that if a product of n polynomials is divisible by an irreducible polynomial p(x) then at least one of them is divisible by p(x). You can assume without a proof that this fact is true for two polynomials.
b) Give an example of three polynomials a(x), b(x) and c(x), such that c(x) divides a(x) ·b(x), but c(x) does not divide neither a(x) nor b(x).

Show that, given any 3 integers, at least 2 of them must have the property that their difference is even.

Post your answer to one (1) of the following questions:

• What are the three steps for solving a quadratic equation?
• When do you have to use the quadratic formula?
• When can you use the root method to solve a quadratic equation?

Directly and completely answer the question(s).

Clearly and accurately explain your answer based on factual information.

Include examples, illustrations and/or applications in your answer(s).

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