An introduction to ordinary differential equations James C.Robinson 대학교재솔루션
An introduction to ordinary differential equations 솔루션 James C.Robinson저 2004년판
An Introduction to Ordinary Di?erential Equations Exercises and Solutions
James C. Robinson
1 Radioactive decay and carbon dating
Exercise 1.1 Radioactive isotopes decay at random, with a ?xed probability of decay per unit time. Over a time interval ?t, suppose that the probability of any one isotope decaying is k?t. If there are N isotopes, how many will decay on average over a time interval ?t? Deduce that N (t + ?t) ? N (t) ? ?N k?t, and hence that dN/dt = ?kN is an appropriate model for radioactive decay. Over a time interval ?t, N k?t isotopes will decay. We then have N (t + ?t) ? N (t) = ?N k?t. Dividing by ?t gives N (t + ?t) ? N (t) = ?N k, ?t and letting ?t → 0 we obtain, using the de?nition of the derivative, dN = ?kN. dt Exercise 1.2 Plutonium 239, virtually non-existent in nature, is one of the radioactive materials used in the production of nuclear weapons, and is a by-product of the generation of power in a nuclear reactor. Its half-life is approximately 24 000 ye
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