Computer Project 1
Math 2250-002, Summer 2010
The following exercises are designed to help you develop a familiarity with
common graphical tools used to understand solution curves of di erential equations. In particular, you will need to use the program d eld to plot representative solution curves to a variety of di erential equations, many of which can
not be easily solved explicitly. A Java implementation of d eld can be found at
the web page http://math.rice.edu/d eld/dfpp.html.
For each of the following exercises, it will be necessary for you to provide
printouts that demonstrate that you have used d eld as required. On a Mac,
one way to do this is to take a screenshot by pressing command+shift+4. The
mouse arrow will then become a crosshair, and you can select a section of the
screen that will be saved as a .png le on your desktop.
1. Consider the di erential equation
dy
dx
= 5y 6e x:
(1)
(a) Use d eld to plot several ( 15) solution curves to this di erential
equation in the window f0 x 3; 1 y 3g.
(b) Notice that limx!1 y(x) = 1 or limx!1 y(x) = 1 for most solutions of this di erential equation. Because this is a rst-order linear
di erential equation, we can solve it explicitly. Do so, and determine
what particular solutions of this di erential equation, if any, are such
that limx!1 y(x) = c, where 1 < c < 1.
(c) Why is the behavior of the solutions you identi ed in part (b) so
di erent from the other, more characteristic, solutions?
2. Consider the di erential equation
dy
dx
= sin(x y):
(2)
(a) Use d eld to plot several solution curves to this di erential equation.
Use this plot to formulate a qualitative description of the behavior
of these solution curves.
(b) Observe that there appear to be straight-line solutions If this is not
apparent from your plot, try using a larger window ( 10 x 10,
10 y 10 should do nicely). This suggests that y(x) = ax + b is
a solution for some values of a and b. Substitute y = ax + b into the
di erential equation to determine what the coecients a and b must
be to get a solution.
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(c) A general solution for this di erential equation is given by
x 2 C y (x) = x 2 tan 1
:
x
C
(3)
Find a value for C that corresponds to a straight-line solution of this
equation.
(d) Notice that the C you found in part (c) does not give the solution
y (x) = x (=2). In fact, no value of C corresponds to this solution.
Why is this so? Are there any values of C for which the corresponding
solution curves lie close to this straight-line solution curve?
3. Consider the di erential equation
dy
dx
= r y2 :
(4)
(a) Use d eld to plot several solution curves for this di erential equation
for r = 1. Use this plot to formulate a qualitative description of the
behavior of these solution curves.
(b) Now plot several solution curves for r = 1. How does the behavior
of solutions di er in this case than those in part (a)?
(c) Describe are there other values for r that give rise to solutions whose
behavior is qualitatively di erent than those in parts (a) or (b)? Plot
several solution curves for any such values of r that you can nd.
(d) Consider now the di erential equation
dy
dx
= ry y2 :
(5)
How does the qualitative behavior of solution curves change as the
value of the parameter r is varied? Provide a representative collection
of solution-curve plots to justify your answer.
4. The di erential equation
2t dy
(6)
= ky(M y) h sin
dt
P
models a logistic population that is periodically harvested and restocked
with period P and maximal harvesting/restocking rate h.
(a) Use d eld to plot several solution curves for this di erential equation
for k = M = h = P = 1. What do these plots suggest about the
qualitative behavior of the population size over time? Compare the
behavior of this population to that of a logistic population without
harvesting or restocking.
(b) How will varying the parameters k, M , h, and P change the behavior
of the solution curves? Include a representative collection of solutioncurve plots to compliment your analysis.
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