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# The presence of toxins in a certain medium destroys a strain of bacteria at a rate

ISBN: 9780387908069 381

## Solution for problem 18 Chapter 1.8

Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition

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Problem 18

The presence of toxins in a certain medium destroys a strain of bacteria at a rate jointly proportional to the number of bacteria present and to the amount of toxin. Call the constant of proportionality a. If there were no toxins present, the bacteria would grow at a rate proportional to the amount present. Call this constant of proportionality b. Assume that the amount T of toxin is increasing at a constant rate c, that is, dT/dt = c, and that the production of toxins begins at time t =O. Let y(t) denote the number of living bacteria present at time t. (a) Find a first-order differential equation satisfied by y(t). (b) Solve this differential equation to obtain y(t). What happens to y(t) as t approaches oo?

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