\begin{problem}{MATH 294}{FALL 1997}{PRELIM II}{?}{}
\prob{Consider the linear difference equation \[ y_{k+3} - 2
y_{k+2} - y_{k+1} + 2 y_k = 0. \]} \probpart{What is the
dimension of the solution set of this equation?} \probpart{Find a
basis for this subspace of $S$.} \probpart{Suppose $u = \{u_k\}$
is a solution to this difference equation where $u_0 = 1, u_1 =
0,$ and $u_4 = 4$. Find a formula for $u_k$. (Hint: Use a linear
combination of the basis vectors that you found in part (b)
above).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1998}{PRELIM 3}{2}{}
\prob{Consider the difference equation \[ y_{k+2} + 4 y_{k+1} +
y_k = 0
\] for $k = 1,2,...,N-2$} \probpart{Find its general solution.}
\probpart{Find the particular solution that satisfies the
boundary conditions $y_1 = 5000$ and $y_N=0$.} \probpartnn{(The
answer involves N.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{98}}{\pr{3}}{4}{}
\prob{The three "spaces" on the simple board game shown are
labeled "C", "I", and "D" for coin, tetrahedron, and dice. On one
turn a player advances clockwise a random number of spaces as
determined by shaking and dropping the object on their present
space (From the C position a player moves 1 or 2 spaces with
equal probabilities, from the $T$ space a player moves 1-4 spaces
with equal probabilities, and from the $D$ space a player moves
1-6 spaces with equal probabilities.). } \probnn{In very long
game what function of the moves end up on the $D$ space on
average? [Hint: Use exact arithmetic rather than truncated
decimal representations.]$$\centerline{\epsfxsize=2.4in
\epsfbox{2_6_112.jpg}}$$}
%------------------------------------------------------------------------------------------------------ picture p. 122
\end{problem}