Find a particular solution for

?^2?′′ + ??′ − 4? = ?^3

Given the fact that the general homogeneous solution is ??(?) = ?1(?^2) + ?2t(?^−2)

Study the roots of the nonlinear equation f(x) = cos(x) + (1 /(1 + e^2x)) both theoretically and numerically. (a) Plot f(x) on the interval x ∈ [−15, 15] and describe the overall behaviour of the function as well as the number and location of its roots. Use the “zoom” feature of Matlab’s plotting window (or change the axis limits) in order to ensure that you are identifying all roots – you may have to increase your plotting point density in order to see sufficient detail!

Please provide MATLAB code as well.

13.7 Table EX 13.7 shows the precedence relationships among the activities to complete a project.

Table EX 13.7

1. Construct an activity on node network for the project.
2. Identify the paths and path project durations.
3. Determine the critical path and the expected project completion time.
4. Find the ES, LS, EF, LF, and slack time for each activity.

6. Passwords are composed from lower- and uppercase letters of the English alphabet,digits and 34 special characters. What is the exponential generating function of the sequence an=number of passwords with at least one capital letter, one number and one special character.

Use the Table button in the Rich-Text Editor to provide an adjacency matrix for a simple graph that meets the following requirements:

1) has 5 vertices
2) is maximal planar

NOTE- If it is true, you need to prove it and If it is false, give a counterexample

f : [a, b] → R is continuous and in the open interval (a,b) differentiable.

a) If f(a) ≥ f(b), then exists a ξ ∈ (a,b) with f′(ξ) ≤ 0. (TRUE or FALSE?)

b) If f is reversable, then f −1 differentiable. (TRUE or FALSE?)

c) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?)

Suppose we modified the Pollard rho method, so the iteration would be f(x) = x^2 mod n instead of f(x) = x^2 + 2 mod n. How well would it perform, and why?

The price of a European call that expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 is expected in two months and again in five months. Risk-free interest rates (all maturities) are 10%. What is the price of a European put option that expires in six months and has a strike price of $30?

The price of an American call on a non-dividend-paying stock is $4. The stock price is $31, the strike price is $30, and the expiration date is in three months. The risk-free interest rate is 8%. Derive upper and lower bounds for the price of an American put on the same stock with the same strike price and expiration date. What are the percentage changes in the values of the two portfolios for a 5% per annum increase in yields?

Calculus 2 question. Please explain clearly.

A bank account earns 5% interest compounded monthly. Suppose that $1,000 is initially deposited into the account, but that $10 is withdrawn each month.

a) Show that the amount in the account after n months is A_n=(1+0.05/12)A_n-1-10; A₀=1000

b) How much money will be in the account after 1year?

c) Suppose that instead of $10, a fixed amount of d dollars is withdrawn each month from the account. Find a value of d such that the amount in the account after each month remains at $1,000

Sn = (1+(1/n))^n

(a) Prove Sn is strictly increasing (b) bounded below by 2 and above by 3

(c) Sn converges to e

(d) Obtain an expression for e

(e) Prove e is irrational

1- Number and describe at least five (5) factors that determine your credit score ?

2-Explain why having a strong credit score is critically important?

1. Let G be the symmetry group of a square and let H be the subgroup generated by a rotation by 180 degrees. Find all left H-cosets.

9. For the differential equation: ?′′ + 2?′ + 5? = ?(?)

(a) Write down the auxiliary polynomial associated to this differential equation and find the roots of the auxiliary polynomial.

(b) Find a basis for the real solution set of the associated homogeneous differential equation ?′′ + 2?′ + 5? = 0.

(c) For each ?(?) that follows, write down the trial particular solution to ?′′ + 2?′ + 5? = ?(?), with undetermined coefficients. Do NOT solve the constants that arise in your trial solution.

(i) ?(?) = 120?^−?

(ii) ??(??) = 68 cos(2?)

(iii) ??(??) = 3?

(iv) ??(??) = 75?^−? sin(2?)

(d) Find a particular solution to ?′′ + 2?′ + 5? = 100 cos(2?) (show your work)

Write the following regarding solutions of a system
→ x′= A → x: 1. The definition of a Fundamental Matrix Φ(t), and apply it in one example of your choosing (for a constant matrix A that you pick). 2. On your example in part 1. calculate the function y(t) = detΦ(t) (the determinant of Φ(t) and check that y(t) solves the first order ODE y′(t) = (trA)y(t) (the trace (trA) is the sum of the entries on the main diagonal of A). 3. Choose a constant 2×2 matrix A different from the one used before. By solving the ODE y′(t) = (trA)y(t) conclude that y(t) is either identically zero, or is never zero.

If F is a field and ?(?),?(?),h(?) ∈ ?[?]; and h(?) ≠ 0.

a) Prove that [?(?)] = [?(?)] if and only if ?(?) ≡ ?(?)(???( (h(?)).

b) Prove that congruence classes modulo h(?) are either disjoint or identical.