Find a general solution to the following​ higher-order equation: y''''+17y''+16y=0

I know you have to rewrite this question into :

r⁴ + 17r² +16 = 0

factoring I get (r² +1)(r²+16), facotring further (r+i)²(r+4i)²

Am I correct so far? and if so how do I approach from this point onwards?

Prove that Kruskal’s algorithm finds a minimum weight spanning tree.

1. For each of the phenomena described below, propose a probability distribution for the numerical variable(s) involved and give the corresponding formula.
e) Compute the probability of observing exactly k alternations of colours in a drawing of 6 balls, with replacement, from an urn containing an equal number of red and blue balls. (Examples: RBBBBB is one alternation, whereas RBBRBB and RRBRBB each have three alternations.)

1.

How many permutations of the 26 letters are there that contain none of the sequences ROCK, STONE, PLUG, FIT or HAY?

5. For each set below, say whether it is finite, countably infinite, or uncountable. Justify your answer in each case, giving a brief reason rather than an actual proof.

a. The points along the circumference of a unit circle.

(Uncountable because across the unit circle because points are one-to-one correspondence to real numbers) so they are uncountable

b. The carbon atoms in a single page of the textbook.

("Finite", since we are able to count the number of atoms in a single page of textbook)(The single page is the limit and it contains a number of carbon atom elements)

c. The different angles that could be formed when two lines intersect (e.g. 30 degrees, 45 degrees, 359.89 degrees, etc….)

("uncountable" because, the different angles that can be made would be in radial from 0-2pi, pi is an example of irrational angle and cannot be counted in the set.

d. All irrationals which are exact square roots of a natural number.

("countably infinite", there are infinite perfect squares as x approaches infinity, so there are infinite exact square roots for a natural number, which is one to one correspondence and is onto therefore is countable

e. All irrationals of the form a+sqrt(b) where a and b are rational numbers.

(im unsure about this one but i would say "uncountable" because not all a+sqrt(b) will be rational, sqrt(b) would have to be ration for it to be a countable, since irrational numbers are countable

f. The set of all squares that can be drawn within a unit circle.

"Uncountable" it has a one to one correspondence from 0 to 2pi

i wanted to crosscheck n see if this is right

The standard technique for determining cardiac output is the indicator dilution method developed by Hamilton. One end of a small catheter is inserted in the radial artery and the other end is connected to a densitometer which can automatically record the concentration of the dye in the blood. A known amount of dye, 5.6mg is injected rapidly the following data is obtained. Fill in the table using Newton’s interpolation to determine the divided difference coefficients.   

t (s)               2               3               8               10             13             14

C (mg/L)       0.89927   0.85987   1.96637   1.54559   0.17987   0.50699

Show that if there are 100 people of different heights standing in a line, then it is possible to find at least 10 people in the order they stand in the line with increasing heights, or at least 12 people with decreasing heights.

Show that is [(n^2) + 2] and [(n^2) - 2] are both primes, then 3 divides n

Solve both ways:

a) y" -2y' + y = e^2x

b) Solve only by variation of parameters

b) y" -9y = x/(e^3x)

c) y" -2y' + y = (e^x)/(x^4)

d) y" + y = sec^3 x

a) Determine the correct form of the particular solution 

y" + y = sin x

b) Solve IVP: y" + y = e^x + x^3; y(0) = 2, y'(0) = 0

c) Solve IVP: y" + y' -2y = x + sin 2x; y(0) = 1, y'(0) = 0

Assume tat the following permutations written in cycle notation are defined on 4 objects, Compute the following compositions:

a. (123)(45)

b. (2345)(12)

c, (124)(253)

d. (25)(23)(24)

e. (35)(123)(125)

f. (12)(23)(35)(145)

Let A and B be sets of real numbers such that A ⊂ B. Find a relation among inf A, inf B, sup A, and sup B.

Let A and B be sets of real numbers and write C = A ∪ B. Find a relation among sup A, sup B, and sup C.

Let A and B be sets of real numbers and write C = A ∩ B. Find a relation among sup A, sup B, and sup C.

A 12 cm by 25 cm by 36 cm box of cereal is lying on the floor on one of its 25 cm by 36 cm faces. An ant, located at one of the bottom corners of the box, must crawl along the outside of the box to reach the opposite bottom corner. What is the length of the shortest path? (* note: the ant can crawl on any of the 5 exposed faces (not the bottom, which is flush with the ground. It can crawl on any edge or face of the box) Hint: Pythagorean theorem.