Diff Eq: Equilibrium Solution sketching?

In summary, the conversation discusses finding the vertex of a parabola graph representing the equation dy/dt = r(1 - y/K)y. The vertex can be found by considering dy/dt as a function of y and using the knowledge of parabolas from algebra. It is also mentioned that for the equation dy/dt = y(y-1)(y-2), the vertex can only be found using calculus.
  • #1
cdotter
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0

Homework Statement


[itex]\frac{dy}{dt}=r(1-\frac{y}{K})y[/itex]


Homework Equations





The Attempt at a Solution


[itex]0=r(1-\frac{y}{K})y[/itex]
y=0 and K.

Plotting dy/dt vs y, the intercepts would then be (0,0) and (0,K).

The book says "the vertex of the parabola is (K/2, rK/4)." Is this something from algebra that I'm forgetting? How do I know/find this?
 
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  • #2
If you plot the equation dy/dt = r(1 - y/K)y, with dy/dt considered to be a function of y, the graph is a parbola that opens downward. The vertex will be on a vertical line midway between the two y intercepts, namely at y = K/2.

This is the same stuff you learned a while back when you were studying the graphs of parabolas.
 
  • #3
What about something like dy/dt = y(y-1)(y-2). I can easily find the points as y=0, 1, and 2. How do I find the vertices? Is there an easy way with algebra or do I just use maximum/minimum from calculus I?
 
  • #4
I don't believe there are any algebraic techniques to find the local max or min - you'll need to use calculus.
 
  • #5
Thank you Mark44. [URL]http://smiliesftw.com/x/bowdown.gif[/URL]
 
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  • #6
cdotter said:
Thank you Mark44. [PLAIN]http://smiliesftw.com/x/bowdown.gif[/QUOTE] [Broken]
That's a really cool smily!
 
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1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model various natural phenomena and physical systems, such as motion, growth, and chemical reactions.

2. What does an equilibrium solution represent in a differential equation?

An equilibrium solution is a solution to a differential equation in which the derivative of the function is equal to zero. This means that the function is not changing and is at a steady state.

3. How do you sketch the equilibrium solutions of a differential equation?

To sketch the equilibrium solutions of a differential equation, you first need to find the points where the derivative of the function is equal to zero. These points represent the equilibrium solutions. Then, you can plot these points on a graph to visualize the behavior of the function around the equilibrium solutions.

4. What is the significance of equilibrium solutions in real-world applications?

In real-world applications, equilibrium solutions represent stable states in a system. They can be used to predict the long-term behavior of a system and determine whether it will reach a steady state or continue to change over time.

5. Are there different types of equilibrium solutions in differential equations?

Yes, there are two types of equilibrium solutions in differential equations: stable and unstable. A stable equilibrium solution is one where the function returns to the equilibrium point after being disturbed, while an unstable equilibrium solution is one where the function moves away from the equilibrium point after being disturbed.

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