Harold wants to borrow $1,000 for 3 years. The interest rate is 6%. How much interest will he pay over the course of the loan?
1 answer:
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A logistic differential equation can be written as follows:
![\frac{dP}{dt} = rP[1- \frac{P}{K}]](https://tex.z-dn.net/?f=%20%5Cfrac%7BdP%7D%7Bdt%7D%20%3D%20rP%5B1-%20%5Cfrac%7BP%7D%7BK%7D%5D%20)
where r = growth parameter and K = carrying parameter.
In order to write you equation in this form, you have to regroup 2:
![\frac{dP}{dt} = 2P[1- \frac{P}{10000}]](https://tex.z-dn.net/?f=%20%5Cfrac%7BdP%7D%7Bdt%7D%20%3D%202P%5B1-%20%5Cfrac%7BP%7D%7B10000%7D%5D%20)
Therefore, in you case r = 2 and K = 10000
To solve the logistic differential equation you need to solve:
![\int { \frac{1}{[P(1- \frac{P}{K})] } } \, dP = \int {r} \, dt](https://tex.z-dn.net/?f=%20%5Cint%20%7B%20%5Cfrac%7B1%7D%7B%5BP%281-%20%5Cfrac%7BP%7D%7BK%7D%29%5D%20%7D%20%7D%20%5C%2C%20dP%20%3D%20%20%5Cint%20%7Br%7D%20%5C%2C%20dt%20%20)
The soution will be:
P(t) =
where P(0) is the initial population.
In your case, you'll have:
P(t) = <span>
Now you have to calculate the limit of P(t).
We know that
</span>
hence,
<span>
</span><span>
</span>
where r = growth parameter and K = carrying parameter.
In order to write you equation in this form, you have to regroup 2:
Therefore, in you case r = 2 and K = 10000
To solve the logistic differential equation you need to solve:
The soution will be:
P(t) =
where P(0) is the initial population.
In your case, you'll have:
P(t) = <span>
Now you have to calculate the limit of P(t).
We know that
</span>
hence,
</span><span>
</span>