- #1

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I have a value which changes over time, I think I can represent this as f(t). It increases by an amount determined by two other values which also change with t: g(t) and h(t), all multiplied by a constant: A. It decreases by an amount determined by its value at the given time: f(t), multiplied by some constant: B.

So I figured:

[tex]\frac{d}{dt}(f(t)) = A g(t) h(t) - B f(t)[/tex]

So the value of the function at time = t1 is:

[tex]\int_{0}^{t_{1}}\frac{d}{dt}(f(t)) dt = A \int_{0}^{t_{1}} g(t) h(t) dt - B \int_{0}^{t_{1}}f(t) dt[/tex]

This gives f(t1) - f(t), f(t) in this case = 0, and so it gives f(t1).

[tex]f(t1) = A \int_{0}^{t_{1}} g(t) h(t) dt - B \int_{0}^{t_{1}}f(t) dt[/tex]

However, the second component on the RHS says that the integral of f(t) dt from 0 to t1 is required. The value of t at a given time between 0 and t1, say t = a, is given by the equation above, evaluated from 0 to a. Therefore:

[tex]f(t1) = A \int_{0}^{t_{1}} g(t) h(t) dt - B \int_{0}^{t_{1}}(A \int_{0}^{t_{1}} g(t) h(t) dt - B \int_{0}^{t_{1}}f(t) dt) dt[/tex]

Does that even make sense?

I'm stuck! Help!

Thanks in advance!