p 83,2.
There are 3 solutions. So the de does not
have a unique solution. That is, the solutions
cross. Find an equilibrium solution. That is, find y =
C for some constant. Now solve via calculus the ode where a constant
C results. Use the fact the ode passes through (0,0) to find a constant
C. From this, find 2 other solutions. Exhibit one where
y = 0 for x < 0, and y = ? for x >= 0 and find a 3rd solution
also. Examples 2.8 and 2.9 on pp 75-77 are helpful.
p 84, 4
(Hint: Use the uniqueness theorem and ask
yourself if it is possible for 2 solutions to cross).
p 89, 1a, b, c.
In c, use pline from Matlab as a check to
your analytical analysis. I want to see a figure similar to FIGURE
2.52 on p 89. This is similar to the discussion of the cutting of
the trees by Howard, Keeling, and Phillips. Also provide 3 hard copies
as asked in the last part of 1., namely 1(b) using dfield5 from Matlab.
Be sure to click on the various regions to indicate several solutions before
saving each graph.
One needs only enter the command
print -dbmp16m a:\pic1
once a graph appears in Matlab. You will see the light come on
the a drive indicating that it is being saved. Then go to
Word and WordPerfect and insert the file in a blank document and print
it. Of course to save an additional file, a different file name should
be entered such as pic2, etc. If we are lucky, perhaps we will soon have
printing ability in Gilbreath 205 and this will be unnecessary.
5 POINT BONUS
p 90, 3. For the sake of argument, let a = 1, b = 1, and c = 2.
Then the right hand side of the ode is (P - 0)(P - 1)(2 - P). Pline
and dfield5 are helpful. Also in your discussion, if P(0) > b, P(t)
-> ?. Also if 0 < P(0) < b, P(t) -> ?. This implies
what? Which of the following a, b, or c is the threshold for survival?
p 105, 1 (Hints: Study y'' on d. In (f) Just substitute into the differential equation.)
p 105,2 (In part c, a rough pencil and paper sketch is ok. In d, use dfield5 from Matlab. Provide a hard copy of the graph. Read discussion above for obtaining such a hard copy.
p 107, 6 a Be sure to give reasons. Merely using dfield5
from matlab is not enough. I want
an analytical discussion.
p 114, 1
p 125, (Euler's method) 1a iii and iv. I want a strictly
paper and pencil solution with the aid of a calculator for doing the arithmetic
calculations such as multiplication, division, addition, and subtraction
only.