find the general solution

2xy^3+e^x+(3x^2y^2+siny)y'=0

xy'=6y+12x^4y^(2/3)

(2x+1)y'+y=(2x+1)^(3/2)

Intro to statistical inference questions! (I got two exams that I need to focus on, so I need help with my homework. Please provide detail if you can, thank you so much!)

1) Give an example of three events E, F, and G so that each pair of events is mutually exclusive.

2) Consider a situation where #(all) = 100, #(E) = 32, #(F) = 52, and #(E ∩ F) = 13.

i. Find P(E | F).

ii. Calculate #(E ∩ F) / #(F) and explain why this matches the value in part 1.

3) Suppose we have 30 shuffled cards numbered 1-30. What is the probability of drawing an even value given that the value is greater than 9?

4) Suppose we roll a 6-sided die two times. What is the probability of the sum of the values being greater than 7 given that the first roll was a 5?

5) Suppose we flip a coin two times. Show that flipping heads on the 1st flip is independent of flipping heads on the second flip.

6) Suppose we roll a 6-sided die one time. Are the following events independent? E : roll a value divisible by 3 and F : roll a value greater than 3.

The “divide and average” method, an old time method for approximating the square root of any positive number a, can be formulated as

x = (x + a/x) / 2

Write a well-structured M-file function based on the while…break loop structure to implement this algorithm. At each step estimate the error in your approximation as

ε = abs(( Xnew − Xold )/Xnew

Repeat the loop until e is less than or equal to a specified value. Design your program so that it returns both the result and the error. Make sure that it can evaluate the square root of numbers that are equal to and less than zero. For the latter case, display the result as an imaginary number. Test your program by evaluating a = 0, 2, 10 and -4 for ε = 1×10−4. Hint: This is similar to the IterMeth function discussed in class.

Give an example of integers a, b, m such that a 2 ≡ b 2 (mod m), but a 6≡ b (mod m)

Prove that a subspace of R is compact if and only if it is closed and bounded.

Choose the general slicing method, the disk/washer method, or the shell method to find the volume of the following solids.

The region bounded by the curves y=x+1, y=12/x, and y=1 is revolved about the x-axis. What is the volume of the solid that is generated?

The​ input-output matrix for a simplified economy with just four sectors​ (natural resources,​ manufacturing, trade and​ services, and personal​ consumption) is given below. Suppose the demand​ (in millions of​ dollars) matrix is matrix D given below. Find the amount each sector should produce.

Production levels of

units from natural​ resources,

units from​ manufacturing,

units from trade and​ services, and

units from personal consumption are needed.

​(Round to the nearest whole number as​ needed.)

In the proof of Theorem 4.7 (Euclid’s proof that there are infinitely many primes), the argument uses calculation of a number N. In each case below, suppose for the sake of demonstrating a contradiction, that the given list is the entire list of prime numbers. Calculate N and then factor N into primes to see that you do get a contradiction.

(a) 2, 3, 5, 7, 11

(b) 2, 3, 5, 7, 11, 13, 17, 19

(c) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

You are developing primers for a wildlife forensic case and want to identify both species and distinguish between individuals. a) For which type of analysis would you want to develop/use degenerate primers? Why? b) If the primer you developed for species ID is 14bp, the mitochondrial genome is 18500 bp, and the nuclear genome is 3.2x109 bp, how many times would you expect it to bind to each of the respective genomes assuming the same primer could bind to both the mtDNA and nDNA genomes? (2marks) c) Comment on the specificity of the primer if this primer was intended to amplify only a mtDNA region and not a nDNA region.

1) In this problem, you may use the fact (which we will prove in Chapter 6) that an integer n is not divisible by 3 if and only if there exists an integer k such that n = 3k + 1 or n = 3k + 2.

(a) Prove that for all integers n, if 3 | n2, then 3 | n.

2) Let a and b be positive integers. Prove that if a | b and b | a, then a = b.

3) Determine whether each statement is true or false. If true, then prove it. If false, then provide a counterexample.

(a) The sum of two irrational numbers is irrational.

(c) The product of a nonzero rational number and an irrational number is irrational.

Solve the following differential equations using Taylor series centered at 0. It’s enough to find the recurrence relation and the first 3 terms of the series.

(a) y''− 2y' + y = 0

(b) y'' + xy' + 2y = 0

(c) (2 + x^2 )y'' − xy'+ 4y = 0

1: Let X be the set of all ordered triples of 0’s and 1’s. Show that X consists of 8 elements and that a metric d on X can be defined by ∀x,y ∈ X: d(x,y) := Number of places where x and y have different entries.

2: Show that the non-negativity of a metric can be deduced from only Axioms (M2), (M3), and (M4).

3: Let (X,d) be a metric space. Show that another metric D on X can be defined by ∀x,y ∈ X: D(x,y) := d(x,y)/(1 + d(x,y)).

4: Let (X,d) be a metric space.

1. Show that every open d-ball is a d-open subset of X.
2. Show that every closed d-ball is a d-closed subset of X.

5: Let (X,d) be a metric space. Show that a subset A of X is d-open if and only if it is the union of a (possibly empty) set of open d-balls.

Let X be a subset of R^n. Prove that the following are equivalent:

1) X is open in R^n with the Euclidean metric d(x,y) = sqrt((x1 - y1)^2+(x2 - y2)^2+...+(xn - yn)^2)

2) X is open in R^n with the taxicab metric d(x,y)= |x1 - y1|+|x2 - y2|+...+|xn - yn|

3) X is open in R^n with the square metric d(x,y)= max{|x1 - y1|,|x2 - y2|,...,|xn -y n|}

(This can be proved by showing the 1 implies 2 implies 3)

(TOPOLOGY)