An introduction to the theory of equations by Florian Cajori

By Florian Cajori

Initially released in 1904. This quantity from the Cornell collage Library's print collections used to be scanned on an APT BookScan and switched over to JPG 2000 structure by means of Kirtas applied sciences. All titles scanned hide to hide and pages might comprise marks notations and different marginalia found in the unique quantity.

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Solution. Multiplying both sides of the differential equation by y gives ydyldt= -t. Hence (12) Now, the curves (12) are closed, and we cannot solve for y as a single-valued function of t. The reason for this difficulty, of course, is that the differential equation is not defined wheny = 0. Nevertheless, the circles t 2 + y 2 = c 2 are perfectly weil defined, even when y = 0. Thus, we will call the circles t 2 + y 2 = c 2 so/ution curves of the differential equation dy I dt = - t 1y. More generally, we will say that any curve defined by (7) is a solution curve of (4).

18 to 140. This implies thaty 0 will also vary over a very large interval, since the number of disintegrations of lead-210 is proportional to the amount present. 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.

5 Population models This is a linear equation and is known as the Malthusian law of population growth. If the population of the given species is p 0 at time t0 , then p(t) satisfies the initial-value problern dp(t)/ dt = ap(t), p(t0 ) = p 0 • The solution of this initial-value problern is p(t) = p 0 ea(t-tol. Hence any species satisfying the Malthusian law of population growth grows exponentially with time. Now, we have just formulated a very simple model for population growth; so simple, in fact, that we have been able to solve it completely in a few lines.

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