Find the general power series solution for the differential equation
2x^2y''-xy'+(x^2 +1) y =0 about x=0

(Answer should be given to the x^4+r term)

A tank initially contains 10 liters of clean water (with no salt in it). Every minute, a solution containing one liter of water and one gram of salt is added to the tank, and two liters of fluid are drained from the tank. The fluid in the tank is mixed continuously so that the salt is distributed evenly throughout the water in the tank. How many minutes does it take for the tank to contain 2 grams of salt?

Prove or disprove: In a hyperbolic geometry, if any right triangle has its hypoteneuse and one leg congruent (respectively) to the hypoteneuse and one leg of another right triangle, then the two right triangles are congruent.

Find and solve a recurrence relation for the number of ways to stack n poker

chips using red, white and blue chips such that no two red chips are together.

Use your solution to compute the number of ways to stack 15 poker chips.

We say that an infinite sequence a0,a1,a2,a3,… of real numbers has the limit L if for every strictly positive number ε, there is a natural number n such that all the elements an,an+1,an+2,… are within distance ε of the value L. In this case, we write lim a = L.

1. Express the condition that lim a = L as a formula of predicate logic. Your formula may use typical mathematical functions like + and absolute value and mathematical relations like > or ∈ . Check that your formula has exactly two free variables: a and L.
   

2. For any two infinite sequences of real numbers a = a0,a1,a2,… and b = b0,b1,b2,… we define their sum (written a+b) to be an infinite sequence c0,c1,c2,c3,… such that ci = ai + bi for every i≥0. Prove (in complete sentences of “mathematical English”) that if we have two infinite sequences with lim a = L and lim b = M then lim (a+b) = (L+M).
   

3. Identify 5 places in your Part B proof where you took a step corresponding to a natural deduction rule, explicitly or implicitly. Each place should correspond to a different natural deduction rule.
   

4. Your proof almost certainly used well-known mathematical “facts” about natural numbers or real numbers. These “facts” are just small theorems that are so obvious or well-known that they can be used in proofs without comment. Give predicate logic formulas to specify three theorems you used. (no proofs for these, but they should all be true formulas about numbers, with no free variables.)

given the initial simplex tableau (Matrix):

x              y              s1            s2            p
6              9              1              0              0              300
5              4              0              1              0              180
-3            -4            0              0              1              0

Show the matrices produced by each pivot

Use Newton's method to find all solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. (Enter your answers as a comma-separated list.)

−2x⁷ − 4x⁴ + 8x³ + 6 = 0

for unit quaternions used to represent rotations for 3D prove that :

a) quaternion conjugate corresponds to rotation inverse ( Approach 1: negate the angle in the formula for converting from the axis-angle form and apply trigonometric identities. Approach 2: convincingly discusses what negating and axis description does in terms of rotation effects and in terms of the formula)

b) negated quaternions represent the same rotation ( hint: add 2PI to the rotation angle in the axis angle formula, explain what that means in terms of orientation, and apply trigonometric identities

Consider the root of function f(x) = x ³ − 2x − 5. The function can be rearranged in the form x = g(x) in the following three ways: (a) x = g(x) = x³ − x − 5 (b) x = g(x) = (x ³ − 5)/2 (c) x = g(x) = thirdroot(2x + 5) For each form, apply fixed-point method with an initial guess x0 = 0.5 to approximate the root. Use the error tolerance = 10^-5 to terminate the iteration. If the iteration does not converge within 15 steps, stop and provide an explanation for the divergence. Keep at least 6 significant digits in your calculation.

Find a basis for the subspace of R4 spanned by (1,0,-2,1), (2,-1,2,1), (1,1,1,1), (0,1,0,1), (0,1,1,0) containing the first and fifth vectors

Recall that a∈Z/nZ is called a primitive root modulon, if the order of a in Z/nZ is equal to φ(n). We have seen in class that, if p is a prime, then we can always find primitive roots modulop.Find all elements of (Z/11Z)∗ that are primitive roots modulo 11.

Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be one to one and onto functions. Then g ◦ f : X → Z is one to one and onto; and (g ◦ f)⁻¹ = f⁻¹ ◦ g⁻¹ ).

Structural Induction on WFF For a formula α ∈ WFF we let `(α) denote the number of symbols in α that are left brackets ‘(’, let v(α) the number of variable symbols, and c(α) the number of symbols that are the corner symbol ‘¬’. For example in ((p1 → p2) ∧ ((¬p1) → p2)) we have `(α) = 4, v(α) = 4 and c(α) = 1. Prove by induction that he following property holds for all well formed formulas: • `(α) = v(α) + c(α) − 1

1. For 25 randomly selected students, (

a) how many ways do all the 25 students have different birthday?

b) find the probability that at least two students have the same birthday.

c) find the probability that only two share the same birthday.

d) for probability computed above identify the sample space, the experiment and outcomes.

e) can a random variable be used for the case above? If yes list the possible values for the variable.

I figured out a through c but i need D and E answered please!!

Show that if (1) F₁ and F₂ are connected sets, and (2) F₁ ∩ F₂ is not empty, then  F₁ ∪ F₂ is connected.

also

Suppose that F is connected. Show that F¯ (the closure of F) is also connected.