Slope fields:Can y converge on two roots?

In summary, for simple differential equations with two roots, it is not possible for y to approach both roots simultaneously. One root will diverge while the other converges, depending on the initial condition. In the case of multiple roots, it is possible for y to converge on one root while the others diverge, but the specific outcome will depend on the initial condition. It is not possible for y to converge on multiple roots simultaneously.
  • #1
craighenn
5
0
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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  • #2
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.
 

FAQ: Slope fields:Can y converge on two roots?

Can a slope field have multiple values for the same x-coordinate?

Yes, a slope field can have multiple values for the same x-coordinate. This is because different points on a curve can have different slopes and therefore, different values in the slope field.

How do you interpret the slope field of a function?

To interpret the slope field of a function, you look at the direction and steepness of the lines. The direction of the line represents the slope of the function at that point, while the steepness indicates the magnitude of the slope.

Can the slope field of a function have a negative slope?

Yes, the slope field of a function can have a negative slope. This can occur when the function is decreasing at a certain point, resulting in a negative slope value in the slope field.

How does the slope field of a function relate to its graph?

The slope field of a function is related to its graph in that the direction of the lines in the slope field represents the direction of the curve at that point. The steepness of the lines also indicates the steepness of the curve at that point.

Can a function have a slope field with a constant slope?

Yes, a function can have a slope field with a constant slope. This can occur when the function is a straight line, where the slope remains the same at every point on the line.

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