A 128-lb weight is attached to a spring having a spring constant of 64 lb/ft. The weight is started in motion with no initial velocity by displacing it 6 in below the equilibrium position and by simultaneously applying an external force F(t) = 8sin(4t). a. Assuming no air resistance, find the equation of motion. b. What is the long term behavior of the motion? Please use differential equations to solve this

A 10- kg mass is attached to a spring, stretching it 0.7 m from its natural length. The mass is started in motion from the equilibrium position with an initial velocity of 1 m/s in the downward direction. a. Find the equation of motion assuming no air resistance. b. When is the first time the mass goes through the equilibrium position after it was set in motion. c. Find the equation of motion assuming air resistance equal to -90x’(t) Newtons. d. What is the long term behavior of the motion described in part (c)? Use differential equations to solve this.

1252) y=(C1)exp(Ax)+(C2)exp(Bx)+F+Gx is the general solution of
the second order linear differential equation:
   (y'') + ( 6y') + (-27y) = ( 2) + ( -3)x. Find A,B,F,G, where
A>B. This exercise may show "+ (-#)" which should be enterered into
the calculator as "-#", and not "+-#". ans:4

3. Let S3 act on the set A={(i,j) : 1≤i,j≤3} by σ((i, j)) = (σ(i), σ(j)).

(a) Describe the orbits of this action.

(b) Show this is a faithful action, i.e. that the permutation represen- tation φ:S3 →SA =S9

(c) For each σ ∈ S3, find the cycle decomposition of φ(σ) in S9.

(a) Prove that Sn is generated by the elements in the set {(i i+1) : 1≤i≤n}.

[Hint: Consider conjugates, for example (2 3) (1 2) (2 3)−1.]

(b) ProvethatSn isgeneratedbythetwoelements(12)and(123...n) for n ≥ 3.

(c) Prove that H = 〈(1 3), (1 2 3 4)〉 is a proper subgroup of S4.

f(x) and g(x) are both functions with Domains and Codomains all positive Reals.

f(x) = 2x² and g(x) = 4x - 1.   If f(g(x) = 8, what is x?

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If f(x) = 7x + 3 and g(x) = 4 - x³, then what is g(f(-2))?

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Let R be the relation from A to B, where A = {1,2,3} and B = { 5,6,7,8}

R = {(1,6), (2, 8), (3, 7), (2,6)}

Remove one ordered pair so that R will be a function from A to B.

2. Use the Laplace transform to solve the initial value problem.
?"+4?=?(?), ?(0)=1, ?′(0)=−1

  = { 1, ? < 1
where ?(?) =   {0, ? > 1.

1. Use the Laplace transform to solve the initial value problem.
?"+4?′+3?=1−?(?−2)−?(?−4)+?(?−6), ?(0)=0, ?′(0)=0

2. Use the Laplace transform to solve the initial value problem.
?"+4?=?(?), ?(0)=1, ?′(0)=−1

    = { 1, ? < 1
where ?(?) =   {0, ? > 1.

Solve the system of linear equations using partial pivoting.

144a1 + 12a2 + a3 = 279.2

64a1 + 8a2 + a3 = 177.2

25a1 + 5a2 + a3 = 106.8

consider the countour C given by |z+1|=1.5 oriented counterclockwise. Evaluate the integral_c (z^5+4z^3+9z^2+1+e^4z)/(z^k); where k= 1,2,3,4,5,6,7. Their shoud be seven answers.

y"+xy=0 center x=0

find two linearly independent solutions centered at x=0

Consider A, B and C, all nxn matrices.
Show that:
1) det(A)=det(A^T)
2) if C was obtained from A by changing the i-th row (column) with the j-th row (column). Show that det(C)=-det(A)
3) det(AB)=det(A)det(B)
4) Let C be a matrix obtained from A by multiplying a row by c ∈ F. Show that det(B)=c · det(A)

Question:

include the MATLAB output and commands used with each problem

Generate two random 10 × 10 matrices with numbers between -10 and 10. This can be done with

>> A = randi([-10,10],10,10)

>> B = randi([-10,10],10,10)

1.     with MATLAB to determi whether A and B are invertible matrices .

2.     If A is invertible, use MATLAB to show that A^-1A = I

3.     Determi whether (AB)^-1 = B^-1A^-1

4.     Determin whether (A^T)^-1 = (A^-1)^T

5.     Determe whether (A³)^-1 = (A^-1)³

6.     Determi whether (A + B)(A - B) = A² - B²

Give an example of a cubic polynomial with zero discriminant (other than x^3=0). Moreover, show that the discriminant is zero when two roots coincide.