TABLE OF CONTENTS Differential equations, both of the ordinary and partial type, arise with extraordinary frequency in all sciences and in engineering, as well as in a host of other disciplines. The present text confines itself to ordinary differential equations (ODEs) but will attempt to cover a broad range of disciplines illustrated with representative examples. To give the reader a flavor of what is to come, we start with a pair of illustrative examples drawn from two different disciplines. These are simple problems leading to ODEs that can be integrated by the elementary method of separation of variables. We will subsequently enter into a more detailed discussion both of the classification of ODEs as well as their solution methods. This will put the topic on a firmer footing.
Malthus' law states that the time rate of change of a population p(t) is proportional to p(t). This holds for many populations as long as they are not too large. A more refined model is the logistic law, given by
For the United States, the following parameter values were established in 1845: a = 0.03 and b = 1.6 10 ?4, with p being measured in millions of people and t in years. Given that the U.S. population in 1850 was 23 million, we are asked to predict the population in 1930, and compare it with the actual value of 123 million.
Integration of equation (2.63) by separation of variables yields
where the...
