A cell's division time conform to exponential distribution with expectation 1/r , what is the corresponding differential equation?

Question

The answer is

$$ dx/dt=r∗x $$

and thus

$$ x(t)=x(0)e^{rt} $$

But how do the first equation comes from the question above?

Answer 1

$$ \int \frac{dx}{x} =\int r . dt $$

Integrating gives:

$$ ln x = rt $$

Take exponentials

$$ x = x_{0} e^{rt} $$

The rate of change of the natural log is the inverse function, that is that the time distribution is the rate of change of growth.

So take the derivative of above

$$\frac{d lnx }{dx} = rt $$

$$ \frac{1}{x} = rt $$

$$ \frac{1}{r} = xt $$

Which means that the distribution of xt is proportional to the change in growth rate.