By Florian Cajori
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The fashionable concept of linear differential platforms dates from the Levinson Theorem of 1948. it's only in additional fresh years, although, following the paintings of Harris and Lutz in 1974-7, that the importance and variety of functions of the concept became liked. This e-book supplies the 1st coherent account of the huge advancements of the final 15 years.
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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.