Consider an mass-spring system with the following IVP for its disagreement y (t) at time t greater than or equal to 0. You may assume it is underdamped.

y" + y' + 5y = 0 , y (0) = -2 , y '(0) = -1

(a) Convert this to a DE system IVP in displacement y and velocity v.

(b) Without using technology or solving the second order DE, make a rough sketch of the system solution on a phase plane. It does not need to be precise, but briefly show and explain the direction you give at the initial value, and the long term behavior as t heads towards infinity

Discuss the applications of PDE's in Engineering, Science and Economics with a detailed example for each. In at least one example, discuss the solution to the PDE and how it has practical implications.

Let L = {aⁱb^j | i ≠ j; i, j ≥ 0}.

Design a CFG and a PDA for this language. Provide a direct design for both CFG and PDA (no conversions from one form to another allowed).

Question 1.​ Answer the following questions by making a truth table. Be sure to explain what feature of the truth table you’ve drawn justifies your answer. (That is, indicate which part, or parts, of the table show what the answer to the question is and why)

a) Is ¬P, P ↔ Q, P → Q a logically consistent set of sentences?

b) Is A ∧ (B ∨ C), ¬((A ∨ B) ∧ C) a logically consistent set of sentences?

c) Does ? → ¬? entail ¬(?∨?)?

d) Does the set ¬(L ∧ M), ¬(M ∧ N) entail ¬M?

If an undamped spring-mass system with a mass that weighs 24 lb and a spring constant 9 lbin is suddenly set in motion at t=0 by an external force of 288cos(4t) lb, determine the position of the mass at any time. Assume that g=32 fts2. Solve for u in feet.

Enclose arguments of functions in parentheses. For example, sin(2x).

1. Determine each of the following set of vectors is linearly independent or dependent.

(a) S1 = {(1, 2, 3),(4, 5, 6),(6, 9, 12)}.

(b) S2 = {(1, 2, 3, 4),(5, 6, 7, 8),(3, 2, 1, 0)}.

(c) S3 = {(1, 2, 3, 4),(5, 6, 7, 8),(9, 10, 11, 12)}

Passwords on an ancient computer are required to be 4-5 characters long and made up of lower case letters only (a..z : there are 26 possibilities). Please answer each of these questions below, giving numerical answers. You should also explain how you got each answer, for full or partial credit.

a) Assume that letters may NOT be repeated, and passwords are 4-5 letters long. How many passwords are possible?

b) Assume that letters MAY be repeated, and passwords are 4-5 letters long. How many passwords are possible?

c) Assuming that letters MAY be repeated, what proportion of the total number of passwords in fact *do* have at least one letter used more than once?

d) Letters may be repeated but the following extra rules exist:

• The password cannot be 4 or 5 identical letters like xxxx or bbbbb
• The password cannot be on a list of 71 common 4-5 letter words (none of them are like xxxx or bbbbb!)

How many passwords are possible now?

Note: since this is a practice test question, the answers can be seen in the question comment after you turn in your quiz. Canvas can't grade "essay" type questions for correctness, so you will get a 0 grade whatever you write.

Explain in detail Multilevel Structural Equations Modelling. Discuss your answer in 1000 words

f(x)= x^3-5sinx-23 use Newton iteration to estimate the root Find A, R, Theoritically and Numerically

Some say that Engineering is not a profession
What is the argument they use to support this position?
(Be brief and precise in your answer)

Draw all possible border pieces of a puzzle, each having a different shape.

Border-pieces are puzzle pieces that have at least one smooth edge.

1. Solve the given third-order differential equation by variation of parameters. y''' − 2y'' − y' + 2y = e^3x