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**Unformatted text preview: **umb to estimate the doubling time. Observe first that ln 2 ≈ .693 ≈ .70 .
Therefore, multiplying the numerator and denominator in (2.10) by 100 we have
t2 = ln 2 × 100
70
≈
.
r × 100
% interest rate For example, if the interest rate is r = .07 = 7 % , the estimate gives 70 / 7 = 10 years for the
doubling time, compared to the more exact value of 9.9. In finance texts, this rule is often
called the Rule of 70.!
Money might seem far removed from biology. Yet the same mathematical arguments that allow us
to analyze the growth of money in the bank can be used to model the growth of populations. In
exercises 18 to 21 we describe how the continuous interest model we have described in this section
can be extended to quantify a variety of common financial transactions. In Chapter 5 we will see
that the same differential equations are related to problems of harvesting and conservation in
population biology.
2.4 Summary dy
of an unknown function
dx
and values of the independent variable x and dependent variable y . The collection of all
solutions to the differential equation is called the general solution. This family of solutions will
usually contain an arbitrary constant C , whose value may be determined by giving additional
information about the unknown function. Quite often this extra information is in the form of an
initial condition, y (0) = y0 , specifying the value of the unknown function at x = 0 .
A differential equation expresses a relation between the derivative 26 2 Differential Equations
Differential equations are solved using integration, but the equation may have to be manipulated
algebraically before this can be done. In a separable differential equation, we can separate the
dependent and independent variables on either side of the equation and solve the equation by
integrating each side. An important example of this type with many applications is the equation
dy
dy
= y or more generally
= ry , where r is a constant. The solution of the latter is given by
dx
dx
y = y0 e rx .
2.5 Exercises
dy
verify that for any constant C the expression...

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