Find a particular solution to the differential equation using the Method of Undetermined Coefficients.

x''(t)-2x'(t)+x(t)=96t^2e^t

8. Find the solutions of the following initial value problems

• x ''' + 6x '' + 12x ' + 8x = 0, x(1) = −2, x' (1) = x '' (1) = 0

• x ^(5) + 4x^ (4) + 5x^ (3) = 0, x(0) = x ' (0) = x '' (0) = 1, x^(3) (0) = x^ (4) (0) =0

2. (a) Find the values of a and b such that the eigenvalues of A = |a 1 are 2 and -5. (b) Find the values of a, b and c such that the eigenvalues of A = | 0 1 0 | 0 0 1 | a b c are 3, -2, and 5.

b 1|

1A) Use surface integral to evaluate the flux of
F(x,y,z) =<x^3,y^3,z^3>
across the cylinder x^2+y^2=1, 0<=z<=2

1B) Use the Divergence Theorem to evaluate the flux of F(x,y,z) =<x^3,y^3,z^3>

across the cylinder x^2+y^2=1, 0<=z<=2

A bacterial population is found to increase 50% every 24 hours. We want to answer the question: why
how much does the population multiply in a week? Establish and solve an initial value problem to express the population of bacteria as a function of time and graph this function.

dx/dt = -5x + y , dy/dt = 4x - 2y

Research will be conducted for a reason or set of reasons. As a result of research actions will generally be taken, although, in some instances research will show that no action needs to be taken. Explain how a community service organisation which is using research results to implement changes to practice can ensure that the changes are made with as little disruption to the workplace as possible. (350 words)

Discuss the roles that Green, Thomson, and Stokes played in discovering these theorems and making them widely known.

Let R=R+. Define:

a+b = ab ; a*b = a^(lnb)

1. Is (R+, +, *) a ring?

2. If so is it commutative?

3. Does it have an identity?

Prove that every natural number is odd or even.

A basis of a vector space V is a maximal linearly independent set of vectors in V . Similarly, one can view it as a minimal spanning set of vectors in V . Prove that any set S ⊆ V spanning a finite-dimensional vector space V contains a basis of V .

what is the code to solve and graph the following function in matlab

dv/dt=80.5(1-e^(.4t))

Write a user-defined MATLAB function that converts real numbers in decimal form to binary form. Name the function b = deciTObina (d), where the input argument d is the number to be converted and the output argument b is a 30-element-long vector with 1s and 0s that represents the number in binary form. The first 15 elements of b store the digits to the left of the decimal point, and the last 15 elements of b store the digits to the right of the decimal point. If more than 15 positions are required in the binary form for the digits to the right of the decimal point, the digits should be chopped. If the number that is entered as d is larger than can be stored in b, the function should display an error message. Use the deciTObina in the Command Window to convert the numbers 85.321, 0.00671, and 3006.42.

USING MATLAB....

a.) Create anonymous functions:

fa(x)=sin(x^2)

fb(x)=sin^2(x)

b.) Evaluate them both at x=1/2 pi

c.)Evaluate them both at x=(0,1,2,...,10)^T

d.) Calculate fa(fb(2)) and fb(fa(2))

Prove that equation have a unique solution through the point (0,1). With the knowledge acquired can you exhibit explicitly that unique solution for a and b?

(a) y' = y - y^2

(b) y' = y^2 - y^3