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{\bf NAME: (print!)}  Katryn Keating
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 {\bf E-Mail address:}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

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MATH 336 (1), Dr. Z. ,  Make up Midterm Exam , Tue., Dec. 2, 2025, 2:35-3:55pm on-line
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Please email completed exam to

ShaloshBEkhad@gmail.com

{\bf FRAME YOUR FINAL ANSWER(S) TO EACH PROBLEM}

{\bf 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
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1. \qquad  (out of 10)    
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2.  \qquad  (out of 10)    
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3.   \qquad (out of 10)    
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4.   \qquad (out of 10)    
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5.   \qquad (out of 10)    
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6.    \qquad (out of 10)    
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7.    \qquad (out of 10)    
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8.    \qquad (out of 10)    
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9.    \qquad (out of 20)    

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tot.: \quad  (out of 100)

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{\bf 1}.  (10 points) Solve explicitly the recurrence equation
$$
x_n=x_{n-1}+6x_{n-2} \quad, \quad
$$
with initial conditions
$$
x_0=2, x_1=1 \quad.
$$



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{\bf 2.} (10 points) 
In a certain species of animals, only one-year-old, two-year-old  and three-year olds are fertile.
The probabilities of a one-year-old, two-year-old, and three-year female to give birth to a new female are $p_1$, $p_2$, $p_3$ respectivley.

Assuming that there were $c_0$ females born at $n=0$, $c_1$ females born at $n=1$ and $c_2$ females born at $n=2$, Set up a recurrence
that will enable you to find the {\bf expected} number of females born at time $n$.

In terms of $c_0,c_1, c_2, p_1,p_2,p_3$, how many females were born at $n=3$?


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{\bf 3}.  (10 points)  Prove that $a_1(n)=2^{5^n}$  satisfies the non-linear recurrence equation
$$
a(n)=a(n-1)^5 \quad .
$$
Is the following constant multiple of the seqeuncec $a_1(n)$, given by $a_2(n)=3\cdot 2^{3^n}$, also a solution? Why?


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{\bf 4.} (10 points) Solve the following recurrence subject to the initla conditions.
$$
a(n)=5a(n-1)-6a(n-2)+1  \quad ; \quad a(0)=2 \quad , \quad a(1)=3 \quad.
$$


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{\bf 5}.  (10 points) (a) Find all the steady-states of the non-linear recurrence
$$
x(n)=k x(n-1) (1-x(n-1)) \quad,
$$
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?

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{\bf 6.} (10 points) Show that $x=4$ is a steady-state of the discrete dynamical system, for any real number $k$,
$$
x(n) = x(n-1) - \frac{(1-x(n-1))\,(4-x(n-1))}{k} \quad,
$$
For what values of the parameter, $k$, is $x=4$ a stable steady-state?
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{\bf 7}.  (10 points) You enter a fair casino (where 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
until  you leave either a winner or loser?

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{\bf 8.} (10 points)  Find the equilibrium point(s) of the continuous dynamical system
$$
x'(t)= 7x(t)-3y(t) \quad, \quad
y'(t)=3x(t)-7y(t) \quad .
$$

Is it stable, semi-stable, or not stable? Explain!
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{\bf 9}.  (20 points) (a) (10 points) Find all the steady-states of the discrete dynamical system
$$
a_1(n+1)=\frac{a_1(n)}{4-a_2(n)}
$$
$$
a_2(n+1)=\frac{a_2(n)}{5-a_1(n)}
$$
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{\bf b} (10 points)  For these steady-states, which of them are stable? unstable? semi-stable? Explain.



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