Download 1. Solve the following second-order equations subject to the

1. Solve the following second-order equations subject to the specified initial conditions:
(a) xt − 5xt−1 + 6xt−2 = 0; x0 = 2, x1 = 5
(b) xt+1 − 5xt + 4xt−1 = 0; x0 = 9, x1 = 33
(c) xt = xt−2 ; x0 = 3, x1 = 5
2. Convert the following two, 1st order linear equations into a single second order equation and find their
general solutions. Indicate the long term behavior of solutions, do xt , yt go to 0, ±∞, or oscillate as
t → ∞?
(a)
xt+1
=
3xt + 2yt
yt+1
= xt + 4yt
(b)
xt+1
= xt + yt
yt+1
=
2yt
3. In the original Fibonacci rabbit problem we did in class, a single pair of immature rabbits is placed
in a field. They mature after one month, and then produce a new pair of rabbits each month. Their
population satisfies the equation
rt+1 = rt + rt−1
with r0 = 1, r1 = 1. Suppose instead one pair of mature rabbits is initially placed in a second field.
How many more rabbits are there after 12 months in the second field than the first?
4. A segmental organism (hypothetical, but think of filamentous algae or fungi, or even branching plants)
grows by adding new segments at intervals of 24 hours in several possible ways (see Figure 1):
• A terminal segment can produce a single daughter with frequency p, thereby elongating its branch.
• A terminal segment can produce a pair of daughters (dichotomous branching) with frequency q.
• A next-to-terminal segment can produce a single daughter (lateral branching) with probability r.
Let an be the number of terminal segments, bn be the number of next-to-terminal segments, and sn
be the total number of segments. Assume that all daughters are terminal segments; that all terminal
segments participate in growth (p + q = 1) and thereby become next-to-terminal segments in a single
generation; and that all next-to-terminal segments are thereby displaced and can no longer branch
after each generation.
(a) Write equations for an , bn , sn .
(b) Show that the equations for an and bn can be combined to give
an+1 − (1 + q)an − ran−1 = 0
and explain the equation in words.
(c) Draw the growth of the organism for the first 3 days for the following choices of p, q, r, and
assuming one initial terminal segment. Find the solution for an . How many terminal segments
will there be after 10 days? What will be the total number of segments?
i. p = 1, q = 0, r = 0
ii. p = 0, q = 1, r = 0
iii. p = 1, q = 0, r = 1
1
Figure 1: A hypothetical segmental organism can grow in one of three ways: (a) by addition of a new segment
at its terminal end; by (b) dichotomous branching, in which two new segments are added at its apex; and by
(c) lateral branching, which occurs at a next-to-terminal segment. Terminal segments are dotted, and next
to terminal segments are marked with x’s.
2