

%finalRachel.tex
%begin macros
%\magnification=\magstephalf
\magnification=\magstep1
\def\g{\bigtriangledown}
\def\L{{\cal L}}
%\baselineskip=14pt
%\parskip=10pt
\def\Tilde{\char126\relax}
\font\eightrm=cmr8 \font\sixrm=cmr6 
\font\eighttt=cmtt8
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\def\S{{\cal S}}
\nopagenumbers
\parindent=0pt
\overfullrule=0in
\def\frac#1#2{{#1 \over #2}}
%end macros

\medskip
{\bf NAME: (print!)}  Rachel Adelman
\bigskip


\qquad
 {\bf E-Mail address:}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

\bigskip
MATH 336 (1), Dr. Z. ,  Final Exam,  Tue., Dec.. 16, 2025, 8:00-11:00 am, BECK 003 (for the class)
<P>

\bigskip

{\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
\medskip
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
\bigskip
1. \qquad  (out of 15)    
\bigskip
2.  \qquad  (out of 15)    
\bigskip
3.   \qquad (out of 15)    
\bigskip
4.   \qquad (out of 15)    
\bigskip
5.   \qquad (out of 15)    
\bigskip
6.    \qquad (out of 15)    
\bigskip
7.    \qquad (out of 15)    
\bigskip
8.    \qquad (out of 15)    
\bigskip
9.    \qquad (out of 15)    
\bigskip
10.    \qquad (out of 15)    
\bigskip
11.    \qquad (out of 15)    
\bigskip
12.    \qquad (out of 15)    
\bigskip
13.    \qquad (out of 20)    
\bigskip


----------------------------
\bigskip

tot.: \quad  (out of 200)

\vfill\eject
This page left blank
\vfill\eject

{\bf 1}.  (15 points) Solve explicitly the recurrence equation
$$
x_n\,= \, 4 x_{n-2} +1 \quad, \quad
$$
with initial conditions
$$
x_0=1, x_1=2 \quad.
$$



\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

{\bf 2.} (15 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 $\frac{1}{2}$, $\frac{2}{3}$, respectivley.

Assuming that there were $10$ females born at $n=0$, $12$ females born at $n=1$ . How many females were born at $n=5$?

\vfill\eject


{\bf 3}.  (15 points)  Write the Maple command to solve the following recurrence
(Do not solve it!)

$$
a(n)=10a(n-1)-21a(n-2)+n^3 \quad, \quad a(0)=5 \quad, \quad a(1)=3
$$

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip




{\bf 4.} (15 points) Solve the following initial value problem 
$$
x''(t)+x(t)=0 \quad, \quad x(0)=1 \quad, \quad x'(0)=0 \quad
$$


\vfill\eject



{\bf 6.} (15 points) Write the Maple command to solve the following intial value problem. Do not solve it
$$
y^{(5)}(t)+y^{(3)}(t)+ y(t)= \sin t \quad, \quad y^{(4)}(0)=1,y^{(3)}(0)=0,y^{(2)}(0)=-1, y'(0)=1, y(0)=3.
$$

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

{\bf 7.} (15 points) Solve the recurrence initial value problem
$$
a(n)=a(n-5)-a(n-6) \quad, 
$$
$$
a(0)=0, a(1)=0,a(2)=0,a(3)=0,a(4)=0,a(5)=0 \quad.
$$

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

{\bf 8}.  (15 points) (a) For which values of $k$ is $x=1$ a stable steady-state of the discrete continuous dynamiical system
$$
x(n)=x(n-1)+ \frac{x(n-1)(1-x(n-1))(2-x(n-1))}{k}
$$

\vfill\eject

{\bf 9}.  (15 points) You enter a fair casino (where the prob. of winning a dollar is $0.48$ and the prob. of losing a dollar is $0.52$), 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? 

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip


{\bf 10.} (15 points)  Find the equilibrium point(s) of the continuous dynamical system
$$
x'(t)= x(t)-y(t) \quad, \quad
y'(t)=-x(t)+y(t) \quad .
$$

Is it stable, semi-stable, or not stable? Explain!
\vfill\eject


{\bf 11}.  (15 points)  Find all the steady-states of the discrete dynamical system
$$
a_1(n+1)=\frac{a_1(n)}{2-a_2(n)}
$$
$$
a_2(n+1)=\frac{a_2(n)}{2-a_3(n)}
$$
$$
a_3(n+1)=\frac{a_3(n)}{4-a_1(n)}
$$


\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

{\bf 12}.  (15 points)  Find all the stable steady-states of the discrete dynamical system
$$
a_1(n+1)=\frac{a_1(n)}{2-a_2(n)}
$$
$$
a_2(n+1)=\frac{a_2(n)}{2-a_3(n)}
$$
$$
a_3(n+1)=\frac{a_3(n)}{4-a_1(n)}
$$


\vfill\eject

{\bf 13.} (20 points) Recall that, in the Hardy-Weinberg model, if the fraction of the population of $AA$ is $u$, the fraction of $Aa$ is $v$ (and hence the fraction of $aa$ is $1-u-v$) then in the next generation it is

$$
[u,v]  \rightarrow [u^2 + v\,u + \frac{v^2}{4}, vu + 2u(1 - u - v) + \frac{v^2}{2} + v\,(1 - u - v)] \quad.
$$

If in this generation $\frac{1}{4}$ of the population is $AA$, $\frac{2}{5}$ and $Aa$. What fraction of the population will be $aa$ in $100$ generations?





\end










