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assignment has six questions and is out of 24 total marks.

(a) Solve the IVP

3xn+1 + 5xn − 2xn−1 = 0, n ≥ 1
x0 = 1, x1 = 2.

(b) Find the general solution to the difference equation

xn = 4xn−1 − 5xn−2 + 2xn−3 + 2n.

(c) Find a particular solution to the difference equation

xn = xn−1 + 6xn−2 + n2n

(d) Answer one of the following two questions.

i. Given the second-order difference equation

Make a conjecture about the behavior of all solutions of Eq.(1).

ii. Consider the third-order difference equation (∆E)

Let the initial conditions x−2, x−1, and x0 be any positive real numbers.

Make a conjecture about the behavior of all positive solutions of Eq.(2).

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