# Autonomous differential Equation

The rate at which a bacteria population multiplies is proportional to

the instantaneous amount of bacteria present at any time. The mathematical model for this dynamics can be formulated as follows:

db/dt = kb

where b is a function in terms of the time t, b(t) is the number of bacteria at the time t, and k is a constant. The general solution of this autonomous dierential equation is

b(t) = b(0)e^(kt)

(a) Given that the initial population size b(0) doubles in two hours, fnd k. (Hint: use the logarithmic function)

(b) Find b(t) in terms of b(0) using the value of k found in (a). Simplify your answer by using the power properties of the exponential

function. Your nal answer should involve neither ln, nor e.

(c) Use (b) to answer the following question: in how many hours

will the bacteria population quadruple, that is, be four times the original population b(0)?

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#### Solution Preview

The rate at which a bacteria population multiplies is proportional to

the instantaneous amount of bacteria present at any time. The mathematical model for this dynamics can be formulated as follows:

db/dt = kb

where b is a function in terms of the time t, b(t) is the number of bacteria at the time t, and k is a constant. The general solution of this autonomous dierential equation is

b(t) = b(0)e^(kt)

(a) Given that the initial population size ...

#### Solution Summary

Autonomous differential equation is solved.

autonomous differential equations

A) Check that z(t) = 1 + sqrt(1 + 2t) is a solution of the autonomous differential equation dz/dt = 1/(z-1) with initial condition z(0) =2

b) Estimate z(4) if z obeys the differential equation dz/dt = 1/(z-1) with initial condition z(0) = 2.

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