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Chapter 8: Q22P (page 399)
Solve the equation for the rate of growth of bacteria if the rate of increase is proportional to the number present but the population is being reduced at a constant rate by the removal of bacteria for experimental purposes
Answer
The rate of growth of bacteria N is .
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Show that for a given forcing frequency , the displacement yand the velocity have their largest amplitude when .
For a given , we have shown in Section 6 that the maximum amplitude of y does not correspond to . Show, however, that the maximum amplitude of for a given does correspond to .
State the corresponding results for an electric circuit in terms of
In Problem 33 to 38, solve the given differential equations by using the principle of superposition [see the solution of equation (6.29)]. For example, in Problem 33, solve three differential equations with right-hand sides equal to the three different brackets. Note that terms with the same exponential factor are kept together; thus, a polynomial of any degree is kept together in one bracket.
Using , find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after , and Example 1.
Find the orthogonal trajectories of each of the following families of curves. In each case, sketch or computer plot several of the given curves and several of their orthogonal trajectories. Be careful to eliminate the constant from for the original curves; this constant takes different values for different curves of the original family, and you want an expression for which is valid for all curves of the family crossed by the orthogonal trajectory you are trying to find. See equations to
Consider an equation for damped forced vibrations (mechanical or electrical) in which the right-hand side is a sum of several forces or emfs of different frequencies. For example, in (6.32) let the right-hand side be ,
Write the solution by the principle of superposition. Suppose, for giventhat we adjust the system so that ; show that the principal term in the solution is then the first one. Thus, the system acts as a "filter" to select vibrations of one frequency from a given set (for example, a radio tuned to one station selects principally the vibrations of the frequency of that station).
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