y'=√2x+y+1

general solution of differential equation

Prove that a continuous function for sections(subrectangles) is integrable in R^N

In the proof of bolzano-weierstrass theorem in R^n on page 56 of "Mathematical Analysis" by Apostol, should the inequality be a/2^(m-2) < r/sqrt(n) or something related to n? a/2^(m-2) < r/2 seems not enough

How is truth found in philosophy and how is that different than in science?

How is truth found in math and how is that different than in science?

Please include the ideas of experimentally based vs. non-experimentally based.

Consider the following linear operator: L u = d^2u/dx^2+ 2 du/dx + u.

Consider the eigenvalue problem L u + λu = 0, x ∈ (0, π), u(0) = 0, u(π) = 0.

(a) Determine the possible eigenvalues and eigenfunctions.

(b) Use an integrating factor to put the eigenvalue problem in Sturm-Liouville form.

(c) What is the appropriate inner product for this system?

Show that, for each value of n, the graph associated with the alcohol C_nH_2n+1OH is a tree (the oxygen vertex has degree 2). Draw the tree corresponding to the molecule C₂H₅OH.

Do the following numbers have a finite binary expansion or not:

0.5, 0.125, 0.015625, 0.0125, 0.05

Construct a ring isomorphism Z2 ⊕Z3 → End(Z2 ⊕Z3)

This is considering the Phi function (The Euler Phi Function)

(a) Explain why φ(m) is always even when m ≥3.

(b) φ(m) is rarely a prime number. Find all numbers m so that φ(m) is prime.

I'm solving for part b. From part a, it looks like there is no prime number for phi of m when it's greater than or equal to 3. Now, I notice that Phi of 4 is 2, Phi of 3 is 2, and Phi of 6 is also 2. It looks like 2 is prime so does that mean m = 3, 4, 6 are all the numbers m in which Phi of m is prime? Please let me know if this is correct or post a solution to what the correct answer is to part b.

The downtime per day for a computing facility has mean 4 hours and standard deviation 0.9 hour. (a) Suppose that we want to compute probabilities about the average daily downtime for a period of 30 days. (i) What assumptions must be true to use the result of the central limit theorem to obtain a valid approximation for probabilities about the average daily downtime? (Select all that apply.) The daily downtimes must have an approximately normal distribution. The number of daily downtimes must be greater than 30. The daily downtimes must have an expected value greater than their variance. The daily downtimes must have the same expected value and variance. The daily downtimes must be independent and identically distributed random variables. (ii) Under the assumptions described in part (i), what is the approximate probability that the average daily downtime for a period of 30 days is between 1 and 5 hours? (Round your answer to four decimal places.) (b) Under the assumptions described in part (a), what is the approximate probability that the total downtime for a period of 30 days is less than 119 hours? (Round your answer to four decimal places.)

A farmer has a mixing tank of capacity 1200 liters which she half-filled with pure, fresh water. She pumps into the tank a concentrated liquid fertilizer (CLF) at rate 3 liters/minute, containing 1/3 ∼ 0.333 kg/liter of nitrate (a salt of nitric acid). In addition, she pours the dry powder fertilizer (DPF, the same chemical as a soluble powder) at rate 1/12 = 0.08333 kg/min in the tank; her aim is to get a right solution concentration 0.12kg/liter for the type of soil she has in her field. Unfortunately, the tank is leaking: when the farmer checks it after 90 min (that is, 1.5 hr) she finds the tank containing only 330 liters of solution. Assume the leak is at the bottom of the tank. Then:

(a) Determine the leakage w in liters/min.

(b) In what time would the tank become empty? Let N(t) be the amount of nitrate in the tank at time t.

(c) Write the InValProblem for N(t) and

(d) solve it, for t in min, N(t) in kg.

(e) Find the amount N(t) and the concentration c(t) of nitrate in the tank at time t = tcheck when the farmer checks the tank. Find out:

(f) how much of the CLF in liters and

(g) how much of the DPF in kg has been pumped/poured in the tank by the time t = tcheck she checks the tank. In absence of leakage: determine

(h) the time t = t ∗ when c(t) would be at the right l evel 0.12 and

(i) the volume V of the liquid solution in the tank at t = t ∗ .

The United States Postal Service (USPS) uses 11-digit serial numbers on its money orders. The first ten digits identify the document, and the last digit is the check digit. The check digit is obtained by a11 = a1 + · · · + a10 (mod 9). For example, one of the money order has serial number 16094004377. (a) The first ten digits of the serial number on a USPS money order are 7306125986. Find the last digit (the check digit). (b) Will this scheme detect all single-digit errors? Prove your statement. (c) Will this scheme detect all transposition error? Why?

Explain the difference between the actual definition of a Riemann Integral of function f on the interval [a,b] and the conclusion of the FTOC Part 2.(Fundamental Theorem of Calculus Part 2)