In Exercises 1–11 a direction field is drawn for the given equation. Sketch some integral curves.
1.
Figure 1.4.1.1
: A direction field for
2.
Figure 1.4.1.2
: A direction field for
3.
Figure 1.4.1.3
: A direction field for
4.
Figure 1.4.1.4
: A direction field for
5.
Figure 1.4.1.5
: A direction field for
6.
Figure 1.4.1.6
: A direction field for
7.
Figure 1.4.1.7
: A direction field for
8.
Figure 1.4.1.8
: A direction field for
9.
Figure 1.4.1.9
: A direction field for
10.
Figure 1.4.1.10
: A direction field for
11.
Figure 1.4.1.11
: A direction field for
In Exercises 12 - 22 construct a direction field and plot some integral curves in the indicated rectangular region.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23. By suitably renaming the constants and dependent variables in the equations
and
discussed in Section 1.2 in connection with Newton’s law of cooling and absorption of glucose in the body, we can write both as
where is a positive constant and is an arbitrary constant. Thus, (A) is of the form (C) with , , and , and (B) is of the form (C) with , , and . We’ll encounter equations of the form (C) in many other applications in Chapter 2.
Choose a positive and an arbitrary . Construct a direction field and plot some integral curves for (C) in a rectangular region of the form
of the -plane. Vary , , and until you discover a common property of all the solutions of (C). Repeat this experiment with various choices of and until you can state this property precisely in terms of and .
24. By suitably renaming the constants and dependent variables in the equations
and
discussed in Section 1.1 in connection with Verhulst’s population model and the spread of an epidemic, we can write both in the form
where and are positive constants. Thus, (A) is of the form (C) with , , and , and (B) is of the form (C) with , , and . In Chapter 2 we’ll encounter equations of the form (C) in other applications..
Choose positive numbers and . Construct a direction field and plot some integral curves for (C) in a rectangular region of the form
of the -plane. Vary and until you discover a common property of all solutions of (C) with . Repeat this experiment with various choices of and until you can state this property precisely in terms of and .
Choose positive numbers and . Construct a direction field and plot some integral curves for (C) in a rectangular region of the form
of the -plane. Vary , , and until you discover a common property of all solutions of (C) with .
You can verify your results later by doing Exercise 2.2.27.
