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





{\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 $n=5$?




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

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






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






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



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



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



{\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? 




{\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!



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




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




{\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?





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