use power series expansion for y''+xy'+3x^2y=0 to find the recursive formula and find the first 6 terms of the general solution

Use a graph or level curves or both to find the local maximum and minimum values and saddle points of the function. Then use calculus to find these values precisely. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = 5xye^{−x2 − y2}

1. Determine the form of a particular solution for the following differential equations. (Do not evaluate the coefficients.)

(a) y'' − y' − 6y = x^2 e^x sin x + (2x^3 − 1)e ^ cos x.

(b) y'' − y' − 6y = (2 − 3x^3 )e^3x .

(c) y'' + 4y' + 4y = x(e^x + e^−x )^2 .

(d) y'' − 2y' + 2y = (x − 1)e^x sin x + x^2 e^−x cos x.

2. Find a general solution to the differential equation y'' − 3y' + 2y = x^2 + 1 − x sin x + xe^2x

Prove that if n is an integer and n^2 is even the n is even.

a) The rectangles in the graph below illustrate a    ?    left endpoint    right endpoint    midpoint     Riemann sum for ?(?)=?29f(x)=x29 on the interval [3,7][3,7].  
The value of this Riemann sum is

equation editor

Equation Editor , and this Riemann sum is an    ?    overestimate of    equal to    underestimate of    there is ambiguity     the area of the region enclosed by ?=?(?)y=f(x), the x-axis, and the vertical lines x = 3 and x = 7.

b) The rectangles in the graph below illustrate a    ?    left endpoint    right endpoint    midpoint    Riemann sum for ?(?)=?29f(x)=x29 on the interval [3,7][3,7].
The value of this Riemann sum is

equation editor

Equation Editor , and this Riemann sum is an    ?    overestimate of    equal to    underestimate of    there is ambiguity     the area of the region enclosed by ?=?(?)y=f(x), the x-axis, and the vertical lines x = 3 and x = 7.

1. Provide a regular expression that describes all bit-strings that length is at least one and at most three.

2. Provide a regular expression that describes all bit strings with odd length.

Consider your SID as map it to the vector . K is the index that denotes the number of digits in your SID. Consider now the vector , where Y denotes your year of birth.

„ Write each of the vectors and tell the dimension n=? of the vector space () each of them belongs to. Is it a matrix? If so, what are its dimensions? Would form a “span” of a vector space? Explain why?

„ Is the vector space   belongs to a subspace of the vector space   belongs to? Prove your decision?

SID is studen id number which is 50802030

Y denotes year of birth which is 1992

^**Please give formulas and step by step instructions**^

A cosmopolitan area was deciding what form of security to use in its subway systems. The city ultimately decided to use metal detectors at the entrance of the subway, but it still must hire security personnel to patrol the area. The town currently uses both city police officers (P) and privately contracted security guards (G). The annual cost of one police officer is $65,000 in wages and benefits, while security guards only cost the city $35,000 per year on average. Each security guard reduces the number of muggings and violent crimes by 2 annually and deters 25 free riders. However, because of their focus on more serious crimes, a police officer only deters 15 free riders annually, but their presence prevents an estimated 6 muggings and violent crimes annually. The city must decrease muggings and violent crime by 250 incidents and reduce the number of free riders by 1,000. The town must have at least 12 police officers dedicated to patrolling subways, and because of a previously existing contract, they must employ at least 20 security guards. The city’s objective is to meet these constraints with the least cost combination. How many city police officers and privately contracted security guards should the city hire?

Let G be a group of order 40. Can G have an element of order 3?

The differential equation of motion of ship is given by M d2x/dt2 + C dx/dt + Kx = F + F2*exp(-2t). t >0.

when M = 4.8, C = 3, K = 5, F = 3.8, F2 = 3, x(t=0)=1 and dx/dt (t=0)= 1.

Find:

a)the order of X(s) and the number of poles & Find all poles of X(s)

b)Determine the stability of system.

Numerical MethodsPlease show all steps with clear hand writing.

Use numerical methods to Prove that the following equations have at leat one solution in the given intervals.

(a) x − (ln x)³ = 0, [5, 7]

(b) 5x cos(πx) − 2x 2 + 3 = 0, [0, 2]

5. Ternary strings. A ternary string is a word from the alphabet {0,1,2}{0,1,2}. For example, 022101022101 is a ternary string of length 6.

(a) Enumerate all ternary strings of length 2.
(b) Generalize: how many ternary strings have length r?
(c) A ternary string of length 6 is to be made. How many ways can we choose how many 0's, 1's, and 2's to include in the string?
(d) A ternary string of length 6 is to have two 0's, a 1, and three 2's. How many strings have this property?
(e) What is the probability that a ternary string of length 6 has two 0's, a 1, and three 2's?

The cycloid has parametric equations x = a(t + sin t), y = a(1 - cos t). Find
the length of the arc from t = 0 to t = pi. [ Hint: 1 + cosA = 2 cos² A/2 ]. and the arc length of a
parametric

. True or false, 2 pts each. If the statement is ever false, circle false as your answer. No work is required, and no partial credit will be given. In each case, assume f is a smooth function (its derivatives of all orders exist and are continuous).

If f has a constraint g = c, assume that g is smooth and that ∇g is never 0.

(a) If f has a maximum at the point (a, b) subject to the constraint g = c, then we must have f(a, b) ≤ c. TRUE FALSE

(b) If A and B are square matrices and AB is defined, then BA must also be defined. TRUE FALSE

(c) The function f(x, y) = x 2 − 2y has a maximum subject to the constraint x + y = 1. TRUE FALSE

(d) If (x0, y0) is the point where f attains its minimum subject to the constraint g(x, y) = c, then ∇f and ∇g must point in opposite directions at (x0, y0). TRUE FALSE

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. In the following, show that y₁ and y₂ satisfy the associated homogeneous equation, and then determine a particular solution of the inhomogeneous equation:

b.) ty" - (t + 1)y' + y = t²e^2t; y₁(t) = 1 + t, y₂(t) = e^t (answer should be: 1/2 (t -1) e^2t + 1/2 + t/2 )

c.) t²y" - 3ty' + 4y = t^5/2; y₁(t) = t², y₂(t) = t²ln(t) (answer should be: 4t^5/2 )