By Brian H. Chirgwin and Charles Plumpton (Auth.)
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Additional info for A Course of Mathematics for Engineers and Scientists. Volume 2, 2nd Edition
G. down a narrow tube, is governed by an ordinary differential equation; the rate at which a gas diffuses is proportional to the gradient of its concentration, dc/dx, down the tube. Diffusion and conduction of heat are closely similar processes and give rise to important partial differential equations when they take place in two or three dimensions. The examples which follow together with Exercises 1:8 give other simple examples of the occurrence of differential equations in a variety of problems.
The temperature d of a cooling liquid is known to decrease at a rate proportional to (ft-a), where a is the constant temperature of the surrounding medium. Show that 6-oc must be proportional to e~"*', where / is the time and & is a positive constant. 44 A COURSE OF MATHEMATICS If the constant temperature of the surrounding medium is 15° and the temperature of the liquid falls from 60° to 45° in 4 minutes, find (i) the temperature after a further 4 minutes, (ii) the time in which the temperature falls from 45° to 30°.
Yn9 . . , xn = x0+nh, . . of x, (These values yn should not be confused with those used on pp. ) If these values are plotted on a graph, the curve obtained is the integral curve through the starting point (x0, y a). The steps of the diagram of Fig. 3 are replaced by some numerical method of determining the new value for y when x is increased by the step h. The basis of our geometrical method was to say that y± = yo + hf(xo, yo)9 y2 = yi + hf(xi, ji), etc. This replaces the actual integral curve at A, Pl9 ...