By Martin Braun

Utilized in undergraduate school rooms around the united states, it is a basically written, rigorous advent to differential equations and their functions. totally comprehensible to scholars who've had three hundred and sixty five days of calculus, this e-book distinguishes itself from different differential equations texts via its attractive program of the subject material to attention-grabbing eventualities. This fourth variation contains previous introductory fabric on bifurcation conception and provides a brand new bankruptcy on Sturm-Liouville boundary worth difficulties. computing device courses in C, Pascal, and Fortran are provided through the textual content to teach readers tips on how to observe differential equations in the direction of quantitative difficulties.

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Additional resources for Differential Equations and Their Applications: An Introduction to Applied Mathematics

Example text

Thus, we cannot use Equation (5) to obtain an accurate, or even a crude estimate, of the age of a painting. 15 1 First-order differential equations Table l. Ore and ore concentrate samples. All disintegration rates are per gram of white lead. E. E. 1 However, we can still use Equation (5) to distinguish between a 17th century painting and a modern forgery. The basis for this statement is the simple observation that if the paint is very old compared to the 22 year half-life of Iead, then the amount of radioactivity from the lead-210 in the paint will be nearly equal to the amount of radioactivity from the radium in the paint.

Remark 2. To derive more accurate models of population growth, we should not consider the population as made up of one homogeneous group of individuals. Rather, we should subdivide it into different age groups. We should also subdivide the population into males and females, since the reproduction rate in a population usually depends more on the number of females than on the number of males. Remark 3. Perhaps the severest criticism leveled at the logistic law of population growth is that some populations have been observed to fluctuate periodically between two values, and any type of fluctuation is ruled out in a logistic curve.

In an exactly analogous manner, we can solve the more general dif- ferential equation (4) where fand g are continuous functions of y and t. This equation, and any other equation which can be put into this form, is said to be separable. To solve (4), we first multiply both sides by f(y) to obtain the equivalent equation dy (5) J(y) dt =g(t). , F(y) = quently, F(y(t))= (6) Jf(y)dy. Conse- Jg(t)dt+c (7) where c is an arbitrary constant of integration, and we solve for y = y(t) from (7) to find the general solution of (4).