The d. e. dy / dx = y tells us that at any point (x, y) on a solution curve, the

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The d. e. dy / dx = y tells us that at any point (x, y) on a solution curve, the slope of the 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 is a solution. In fact, y = Ce' is a solution of the d. e. for every constant C, since y' = Ce' = 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), (In 3. 3). or (pi, 3), the slope equals 3; and so on. In Figure N9- la we sec 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 sec the curve of y = e' with slope segments drawn in as follows:

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