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Additional resources for Differential Equations in Applications (Science for Everyone)
The latter component, as is known, is equal to the product of the angular ve— locity dt)/dt and radius r, that is, d0 "t=*a;· But since v, must be equal to v, we have vt:]/(2v)2-02:]/-3:0. v, r W - V 3 v, which, in turn, can be reduced to a single equation, dr/r = d9/ V3, by excluding the variable t. Solving the last differential equation, we find that r=Ce°/Vg, with C an arbitrary constant. o Ch. 1. Construction of Differential Models 55 If we now allow for the fact that destroyer beginsits motion about pole O starting from the polar axis r at a distance of sz: miles away fromO, thatis,r = 1 at 0 = 0 and r = 3 at 9 =—:n;, we conclude that in the first case ii C = 1 and in the second C = 3e”/V; Thus, to fulfill its mission, the destroyer must move two or six miles along a straight line toward the point where the submarine was discovered and then move in the spiral r = e9/VT or the spiral r = 3e(9+”)/V?
4 Precisely, if we suppose that the bowl has the shape of a surface of revolution, in accordance with the notations used in Figure 4, Eq. (24) yields the following result: 1/ 2% Z —--2 a, <25> ks ]/2g where a = vx ·—-= da:/dt is the projection on the :1: axis of the rate of motion of the water’s free surface, which is constant by hypothesis. Squaring both sides of Eq. (25), we arrive at the equation x = cr", (26) with c = aznz/2gk2s”. The latter means that the sought shape of the water clock is obtained by rotating curve (26) about the 2: axis.
8 Differential Models in Ecology Ecology studies the interaction of man and, in general, living organisms with the environment. The basic object in ecology is the evolution of populations. Below we describe differential models of populations that deal with their reproduction or extinction and with the coexistence of various species of animals in the predator vs. * Let x (t) be the number of individuals in a population at time t. A—B. (30) The problem consists in finding the dependence of A and B on :1:.