James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

594 Chapter 9 Differential Equations
The direction field for this differential equation is shown in Figure 9.
I
6
4
2
FIGURE 9
0 1
2 3 t
(b) It appears from the direction field that all solutions approach the value 5 A, that is,
lim Istd − 5
t l `
(c) It appears that the constant function Istd − 5 is an equilibrium solution. Indeed, we
can verify this directly from the differential equation dIydt − 15 2 3I. If Istd − 5, then
the left side is dIydt − 0 and the right side is 15 2 3s5d − 0.
(d) We use the direction field to sketch the solution curve that passes through s0, 0d, as
shown in red in Figure 10.
I
6
4
2
FIGURE 10
0 1
2 3 t
n
Notice from Figure 9 that the line segments along any horizontal line are parallel.
That is because the independent variable t does not occur on the right side of the equation
I9 − 15 2 3I. In general, a differential equation of the form
y9 − f syd
in which the independent variable is missing from the right side, is called autonomous.
For such an equation, the slopes corresponding to two different points with the same
y-coordinate must be equal. This means that if we know one solution to an autonomous
differential equation, then we can obtain infinitely many others just by shifting the graph
of the known solution to the right or left. In Figure 10 we have shown the solutions that
result from shifting the solution curve of Example 2 one and two time units (namely,
seconds) to the right. They correspond to closing the switch when t − 1 or t − 2.
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