How to scale a logistic equation to become dimensionless?

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In summary, the conversation discusses an equation that needs to be scaled to a dimensionless form using substitutions. The person has tried multiple methods and is still unable to get it into the desired form. They mention expanding and rearranging the equation, as well as substituting suggested values, but have been unsuccessful. They are asked to show their work and it is mentioned that the given substitutions will work once c and d are found through simplification.
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miclectric
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Homework Statement



I have an equation: dN/dt = N(r-a(N-b)^2) where r,a,b>0 are constants and I need to scale it to the dimensionless form dn/dtao=n(1-a(n-1)^2), however, I tried many ways and I am still unable to get it into the form. The question also suggests using n=N/c and tao=t/d as a substitution.


Homework Equations





The Attempt at a Solution


I tried to first expand out the N in the original equation and tried a separation of equation form to get the N's all on one side and dt on the other, but I can't seem to do this. I've also tried to substitute the suggested substitutions, however I can't seem to get it to the form either... :(
 
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  • #2
miclectric said:

The Attempt at a Solution


I tried to first expand out the N in the original equation and tried a separation of equation form to get the N's all on one side and dt on the other, but I can't seem to do this. I've also tried to substitute the suggested substitutions, however I can't seem to get it to the form either... :(
Please show your work then. The given substitutions will work, you just have to find c and d (that will become clear once you simplify the equation).
 

Related to How to scale a logistic equation to become dimensionless?

1. How do I determine the appropriate scaling factor for a logistic equation?

The scaling factor for a logistic equation can be determined by comparing the units of the independent and dependent variables. The scaling factor should be chosen such that when multiplied by the independent variable, it results in a dimensionless quantity.

2. What is the purpose of scaling a logistic equation to become dimensionless?

Scaling a logistic equation to become dimensionless allows for a more general and simplified form of the equation. It also ensures that the results are not affected by changes in the units of the independent variable.

3. Can a logistic equation be scaled to become dimensionless without altering its form?

Yes, a logistic equation can be scaled to become dimensionless without altering its form. This can be achieved by multiplying the independent variable by an appropriate scaling factor, as long as the form of the equation remains unchanged.

4. Are there any limitations to scaling a logistic equation to become dimensionless?

One limitation of scaling a logistic equation to become dimensionless is that it may not accurately represent real-world data. This is because the scaling factor is chosen based on the units of the independent and dependent variables, rather than the actual values of the variables.

5. How does scaling a logistic equation to become dimensionless affect the interpretation of its parameters?

Scaling a logistic equation to become dimensionless does not affect the interpretation of its parameters. The parameters still represent the same concepts and have the same interpretations, but the values may differ due to the change in units caused by scaling.

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