find two power series solutions of 2x^2 y''-xy'+(x+1)y=0 about the center at the point x=0

1. Consider the problem of two polluting sources in the region, each of which generated 10 units of pollution for a total of 20 units released into the environment. The government determined that emissions must be reduced by 12 units across the region to achieve the ”socially desirable level of pollution”. Each firm faces different abatement cost conditions modelled as follows: for Polluter 1, marginal abatement cost is MAC₁ = 2.6Q₁, while the total abatement cost is TAC1 = 1.3(Q1)². For Polluter 2, marginal abatement cost is MAC2 = 0.52Q₂, while the total abatement cost is TAC2 = 0.26(Q2)², where Q₁ is the amount of pollution controlled (abated) by Polluter 1, and Q₂ is the amount of pollution controlled (abated) by Polluter 2.

(a) What is the cost effective abatement allocation across polluting sources? What is the total cost to achieve this goal?

(b) Assume that the government implements the 12-units standard uniformly, requiring each polluter to abate by 6 units. What is the total cost to achieve this goal? Is it more / less than the total cost from part a)? Comment on your findings.

Consider now other possible policies like a tradable emission permits (TEP’s) system and an emission tax as ways to achieve the cap of 8 units of emissions.

(c) Assume that the government imposes emission charge set at $4 for each polluter. Show how each firm responses to tax. Does $4 unit tax achieve the 12-unit abatement standard? If not, is $4 unit tax too high or too low? Discuss.

(d) Assume that the government decides to issue permits rather than impose tax. It issues 8 permits, each of which allows the bearer to emit 1 unit of pollution. The government allocates 4 permits to each polluter.

i. If the permits system does not allow for trading, what would be each firm’s response - cost, abatement required to this allocation?

ii. Assume now that trading is allowed and that two firms agree on the purchase and sale of permit at a price of $8.00. What would be each firm’s response - cost, abatement required, revenue to this price?

iii. Does the outcome from part (ii) represent the cost effective solution? If yes - why? If not, describe what happens next.

find the solution of the given initial value problem

1.   y''+4y=t²+3e^t, y(0) =0, y'(0) =2

2. y''−2y'+y=te^t +4, y(0) =1, y'(0) =1

find the general solution of the given differential equation

1. y''+2y'=3+4sin2t

2. 2y''+3y'+y=t² +3sint

At what point do the curves r1(t) = t, 3 − t, 35 + t2 and r2(s) = 7 − s, s − 4, s2 intersect? (x, y, z) = Find their angle of intersection, θ, correct to the nearest degree. θ = °

Consider G = (Z₁₂, +). Let H = {0, 3, 6, 9}.

a. Show that H is a subgroup of G.

b. Find all the cosets of H in G and denote this set by G/H. [Note: If x ∈ G then H +₁₂ [x]₁₂ = {[h + x]₁₂?? | [h]₁₂ ∈ H} is the coset generated by x.]

c. For H +₁₂ [x]₁₂, H +₁₂ [y]₁₂ ∈ G/H define (H+_12[x]₁₂)⊕(H+₁₂[y]₁₂) by(H+₁₂ [x]₁₂)⊕(H+₁₂ [y]₁₂)=H+₁₂ [x+y]₁₂.

d. Show that ⊕ is well defined and construct the addition table for G/H with the operation ⊕.

find the projection vector of the vector v = (2,3,5) onto the plane z = 2x + 3y -1

Second order Differential equation:

Find the general solution to [ y'' + 6y' +8y = 3e^(-2x) + 2x ] using annihilators method and undetermined coeficients.

*PLEASE SHOW ALL WORK*

Consider a damped, forced mass/spring system. Let t denote time (in seconds) and let x(t) denote the position (in meters) of the mass at time t, with x = 0 corresponding to the equilibrium position. Suppose the mass m = 1 kg, the damping constant c = 3 N·s/m, the spring constant k = 2 N/m, the external force is F (t) = 20 cos(2t), the initial position

x(0) = 1 m, and the initial velocity x′(0) = 2 m/s. a. Find the position function x(t).

b. What part of the solution is the transient part and what part is the steady-state part?

1. For an m x n matrix A, the Column Space of A is a subspace of what vector space?

2. For an m x n matrix A, the Null Space of A is a subspace of what vector space?

Use the method of undetermined coefficients to find the complete solutions of the following differential equations.

d²y/dx² − 3 dy/dx + 2y = 2x² + e^x + 2xe^x + 4e^3x .

Given the curve −→r (t) = <sin³ (t), cos³ (t),sin² (t)> for 0 ≤ t ≤ π/2 find the unit tangent vector, unit normal vector, and the curvature.

Another model for a growth function for a limited population is given by the Gompertz function, which is a solution to the differential equation

dP/dt=cln(K/P)P

where c is a constant and K is the carrying capacity. Answer the following questions

1. Solve the differential equation with a constant c=0.15, carrying capacity K=3000, and initial population P0=1000

Answer: P(t)=?

2. With c=0.15, K=3000, and Po=1000, find limt→∞P(t).

Limit:?