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E-Mail address: MATH 428 (2), Dr. Z. , Exam 2, Wed., Nov. 26, 2025, 12:10-1:30pm, BECK 003 FRAME YOUR FINAL ANSWER(S) TO EACH PROBLEM No Calculators! No books! No Notes! To ensure maximum credit, organize your work neatly and be sure to show all your work. Do not write below this line

1. (out of 10)

2. (out of 10)

3. (out of 10)

4. (out of 10)

5. (out of 10)

6. (out of 10)

7. (out of 10)

8. (out of 10)

9. (out of 20) —————————- tot.: (out of 100)

This page left blank 1. (10 points) Solve explicitly the recurrence equation

x_{n} = x_{n−1} + 6x_{n−2} ,

with initial conditions

x_{0} = 2, x_{1} = 1 .

2. (10 points) In a certain species of animals, only one-year-old, two-year-old are fertile. The probabilities of a one-year-old, two-year-old, female to give birth to a new female are p_{1}, p_{2}, respectivley. Assuming that there were c_{0} females born at n = 0, c_{1} females born at n = 1 Set up a recurrence that will enable you to find the expected number of females born at time n. In terms of c_{0}, c_{1}, p_{1}, p_{2}, how many females were born at n = 4?

3. (10 points) Prove that a_{1}(n) = 2^{3^{n}} satisfies the non-linear recurrence equation

a(n) = a(n − 1)^{3} .

Is the following constant multiple of the seqeuncec a_{1}(n), given by a_{2}(n) = 2 · 2^{3^{n}}, also a solution? Why?

4. (10 points) Solve the following recurrence subject to the initla conditions.

a(n) = 3a(n − 1) − 2a(n − 2) + n ; a(0) = 2 , a(1) = 3 .

5. (10 points) (a) Find all the steady-states of the non-linear recurrence

x(n) = kx(n − 1)(1 − x(n − 1)) ,

where k is a positive number between 0 and 4. Express your answer in terms of k (if needed).

(b) For which values of k is x = 0 the only stable steady-state? For which values of k is the other steady-state stable? For which values of k neither of them are?

6. (10 points) Show that x = 4 is a steady-state of all discrete dynamical systems of the form

x(n) = x(n − 1) − \frac{(1 − x(n − 1)) ∗ (4 − x(n − 1)}{k} ,

where k is a non-zero real number. For what values of the parameter k is x = 4 a stable steady-state?

7. (10 points) A gambler enters a fair casino (with the prob. of winning a dollar is \frac{1}{2} and the prob. of losing a dollar is \frac{1}{2}, and you must exit if you are broke or you have 200 dollars. If right now you have 150 dollars, how likely are you to exit a winner? How long would you expect to be in the casino untik you leave either a winner or loser?

8. (10 points) Find the equilibrium point(s) of the continuous dynamical system

x′(t) = 2x(t) − 3y(t) , y′(t) = 3x(t) − 2y(t) .

Is it stable, semi-stable, or not stable? Explain!

9. (20 points) (a) (10 points)Find all the steady-states of the system

a_{1}(n + 1) = \frac{a_{1}(n)}{3 + a_{2}(n)}

a_{2}(n + 1) = \frac{a_{2}(n)}{2 + a_{1}(n)}

b (10 points) Which of them is a stable steady-state?