Exponential growth with an elimination rate

[nrslmz]

Homework Statement

Viruses are reproducing with rate of k ,in t minutes, the function is:
f(x) = Po x e^(kt)
However there is an elimination rate of a viruses per minute.

Homework Equations

The Attempt at a Solution

We can't say that the new function will be:
f(x) = Po x e^(kt) - ax
because the initial number pf viruses are different for every trial. I used succeeding cells with formulae in Excel to generate these values, and it worked. But is there a way to generalize this situation into a formula?

 

[Defennder]

Which variables are you using? Your function is supposedly a that of variable x, f(x) but yet somehow t appears. It should be $$f(t) = P_0 e^{kt}$$ correct?

When you take the rate at which the viruses die into consideration you need to make clear whether or not they are dying at a constant rate, or one is proportionate to their current size. You usually start off with the differential equation. In this case, we know that $$\frac{df}{dt} = kf - D(t,f)$$, where D(t,f) is the death rate in terms of an unspecified function. You can't use the final exponential function and then modify it to take into account the elimination rate if the death rate makes the logistic assumption for example.

The initial population of viruses doesn't matter as well. Those are the numbers you plug in after you solve the DE to find the unknown constants of integration.

 

[nrslmz]

Yes you are right. I was in a bit of hurry when posted this. The variable should be t. We have just finished integral and started differential equations in school, so I kind of guessed that the problem might be solved that way.
Other than that thank you very much. You somehow manage to help everyone in this forum, very admirable:).