is a solution. In fact, y = Cex is a solution of the d.e. for every constant C, since y ′ = Cex = y.
The d.e. y ′ = y says that, at any point where y = 1, say (0, 1) or (1, 1) or (5, 1), the slope of the
solution curve is 1; at any point where y = 3, say (0, 3), (ln 3,3), or (π, 3), the slope equals 3; and
so on.
In Figure N9–1a we see some small line segments of slope 1 at several points where y = 1, and
some segments of slope 3 at several points where y = 3. In Figure N9–1b we see the curve of y =
ex with slope segments drawn in as follows:
FIGURE N9–1aFIGURE N9–1b
Figure N9–1c is the slope field for the d.e. Slopes at many points are represented by small
segments of the tangents at those points. The small segments approximate the solution curves. If
we start at any point in the slope field and move so that the slope segments are always tangent to
our motion, we will trace a solution curve. The slope field, as mentioned above, closely