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By Brian H. Chirgwin and Charles Plumpton (Auth.)

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Extra resources for A Course of Mathematics for Engineers and Scientists. Volume 2

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R. 32) at right angles is called an orthogonal trajectory of that family. Two families of curves are orthogonal trajectories when every curve of one family cuts every curve of the other family at right angles. Suppose a curve of thefirstfamily passes through the point (x, y) and has slope tan ip (= dy/dx) there. Then the slope of the member of the second (orthogonal) family which passes through (,v, y) is tan y i where tan y> tan \pi = — 1. Hence the differential equation of the ortho- § 1 : 6] FIRST ORDER DIFFERENTIAL EQUATIONS gonal family, obtained by replacing 29 in eqn.

The examples which follow together with Exercises 1:8 give other simple examples of the occurrence of differential equations in a variety of problems. Example 1. The rate of decay of a substance is kx, where x is the amount of the substance remaining and k is a constant. Show that the half-life (the time in which the amount is halved) of the substance is T = (In 2)\k. Find also the time required for five-eighths of the substance to disintegrate, giving the result in terms of the half-life T. The rate of decay of the substance is - dx/d/.

The resistance to the motion of the particle is kv- per unit mass, where v is its velocity and k is a constant. Show that the particle strikes the plane with velocity V where If the particle rebounds from the plane without loss of energy and reaches a maximum height b in the subsequent motion, show that 13. A particle of unit mass moves from rest at time t = 0 along a straight line under a force depending only on its velocity and having the value 3—2v-v 2 when the velocity is v. Prove that in the subsequent motion the velocity is given by where 2 tanh a = 1.

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