Slope fields:Can y converge on two roots?

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SUMMARY

The discussion centers on the behavior of solutions to the differential equation y' = y(y-3), which has two roots at y = 0 and y = 3. It is established that a solution cannot converge to both roots simultaneously; instead, it will diverge from one root while converging to the other based on the initial conditions. Specifically, one root is stable (y = 3) while the other is unstable (y = 0), confirming that limits are unique in this context.

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craighenn
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For simple differential equations like y'=y(y-3) where there 2 roots, is it possible for y to approach y=root for both roots? Or must one diverge while the other converges? For multiple roots, can it only converge on one root, the others all diverging?

If it can converge on multiple roots, can you provide an equation example?
 
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What do you mean "converge to both roots"? The fate of y will depend on the initial condition. Certainly, no initial condition can result in y converging to both 0 and 3, since limits are unique. In the case of your example, one fixed point is stable and the other is unstable.
 

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