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# Autonomous differential Equation

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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 di erential equation is

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

(a) Given that the initial population size b(0) doubles in two hours, f nd 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

(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)?

https://brainmass.com/math/calculus-and-analysis/autonomous-differential-equation-277776

#### 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 di erential equation is

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

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

#### Solution Summary

Autonomous differential equation is solved.

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