Suppose that you follow the size of a population over time. When you plot the size of the population versus time on a semilog plot (i.e., the horizontal axis, representing time, is on a linear scale, whereas the vertical axis, representing the size of the population, is on a logarithmic scale), you find that your data fit a straight line which intercepts the vertical axis at 1 (on the log scale) and has slope -0.43. Find a differential equation that relates the growth rate of the population at time t to the size of the population at time t.

Question

I'm having a hard time digesting this. Help please?? Thank you so much for your time!

Answer 1

Use the point-slope form of a line, but use ln y instead of y

(ln y - 1)/x = -.43

ln y = 1 - .43x

y = exp(1 - .43x)

when x=0, y= exp(1), so ln y = 1
when x = 2.32, y = exp(1-1) = 1, so ln y = 0

Answer 2

Hmmm. Forget the above.

We want y(0) = 1 and y(2.32) = 0.1

So, assuming y = ae^bx we have

1 = ae⁰
a = 1

so, y = e^bx

.1 = e^2.32b
ln .1 = 2.32b
b = -2.3/2.3 = -1

So, y = e^-x

Am I still wrong here?