curve is equal to its y-coordinate. Since the d.e. says that y is a function whose
derivative is also y, we know that
y = exis 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 N9–1a