A forest has three components and nine vertices. How many edges does it have? Explain. It is not enough to give an example; you must show that all examples of such a forest have the asserted number of edges.

The Bessel equation of order p is t²y" + ty' + (t² - p²)y = 0. In this problem, assume that p = 1/2:

a.) Show that y₁ = sin(t / sqrt(t)) and y₂ = cos(t / sqrt(t)) are linearly independent solutions for 0 < t < infinity.

b.) Use the result from part (a), and the preamble in Exercise 3, to find the general solution of t²y" + ty' + (t² - (1/4))y = t^3/2cos(t). (answer should be: 1/2 sin(t) sqrt(t))

**Preamble of Exercise 3: The formula for a particular solution given in (3.42) applies to the more general problem of solving y" + p(t)y' + q(t)y = f(t). In this case, y₁ and y₂ are independent solutions of the associated homogeneous equation y" + p(t)y' + q(t)y = 0.

Please show work!

y''-2y'+1=0; y(0)=0, y'(0)=-1

Solve the given initial-value problem. y′′′ + 18y′′ + 81y′ = 0, y(0) = 0, y′(0) = 1, y′′(0) = −10.

In a spring-mass-dashpot system, a force of 1 Newtons is required to stretch the spring for .05 meters. A mass of 4 kg is hung from the spring and also attached to a viscous damper that has a damping constant 8 Newton-sec/m. The mass is suddenly set in motion from its equilibrium location at t = 0 by an external force of 8 cost Newtons with initial velocity 0 m/sec. Find the transient solution and the steady state solution of the system.

Solve the initial value problem. Use the method of undetermined coefficients when finding a particular solution. y'' + y = 8 sin t; y(0) = 4, y' (0) = 2

2. You’re playing a game of Magic: The Gathering with a deck that consists of 60 cards, 24 of which are land cards. You draw an opening hand of seven cards which contains exactly one land and six non-lands. (For this problem, you are welcome to read and use the tools of this article1, but your answers must include clear explanations of the mathematical reasoning you’re using!)

(a) What is the probability of this event (that your hand of seven cards contains exactly one land)? Explain your answer clearly.

(b) On your first turn you draw a card from the top of your remaining deck (since you have seven cards in hand, 53 remain in the pile you’re drawing from). What is the probability that you draw a land on your first turn? Explain your answer clearly.

(c) What is the probability that you will draw on land on either your first draw or your second draw (or both)? Explain your answer clearly.

Hi! I hope all is well. Please use a method that is the most straightforward, and easiest to understand.

Many thanks!

A nitric acid solution flows at a constant rate of 6L/min into a large tank that initially held 200L of a 0.5% nitric acid solution. The solution inside the tank is kept well-stirred and flows out of the tank at a rate of 8L/min. If the solution entering the tank is 20% nitric acid, determine the volume of nitric acid in the tank after t min. When will the percentage of nitric acid in the tank reach 10%? (Ans: (0.4)(100-t)-(3.9x10-7 )(100-4)4 L; 19.96min)

Problem 3 Find the solutions to the general cubic a x^3 +b x^2+c x +d=0 and the solutions to the general quartic a x^4+b x^3+c x^2+d x+e=0. Remember to put a space between your letters. The solutions to the general quartic goes on for two pages it is a good idea to maximize your page to see it. It is a theorem in modern abstract algebra that there is no solution to the general quintic in terms of radicals.

Please Answer 4.4.2: Do you see any problems with the choice of hash functions in Exercise 4.4.1? What advice could you give someone who was going to use a hash function of the form h ( x ) = ax + b mod 2 k ?

Exercise 4.4.1:

Suppose our stream consists of the integers 3, 1, 4, 1, 5, 9, 2,6, 5. Our hash functions will all be of the form h (x) = ax+b mod 32 for some a and b. You should treat the result as a 5-bit binary integer. Determine the tail length for each stream element and the resulting estimate of the number of distinct elements if the hash function is:

(a) h(x) = 2x+ 1 mod 32.

(b) h(x) = 3x+ 7 mod 32.

(c) h(x) = 4x mod 32.

In this question you will find the intersection of two planes using two different methods.

You are given two planes in parametric form,

1. Find vectors n1 and n2 that are normals to Π1 and Π2 respectively and explain how you can tell without performing any extra calculations that Π1 and Π2 must intersect in a line.
2. Find Cartesian equations for Π1 and Π2.
3. For your first method, assign one of x1, x2 or x3 to be the parameter ω and then use your two Cartesian equations for Π1 and Π2 to express the other two variables in terms of ω and hence write down a parametric vector form of the line of intersection L.
4. For your second method, substitute expressions for x1, x2 and x3 from the parametric form of Π2 into your Cartesian equation for Π1 and hence find a parametric vector form of the line of intersection L.
5. If your parametric forms in parts (c) and (d) are different, check that they represent the same line. If your parametric forms in parts (c) and (d) are the same, explain how they could have been different while still describing the same line.
6. Find m=n1×n2 and show that m is parallel to the line you found in parts (c) and (d).
7. Give a geometric explanation of the result in part (f)