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
42 A COURSE OF MATHEMATICS The information given can be expressed in the two equations where kx is a constant of proportionality. ltf, Exercises 1 : 8 1. The pressures at height h and at ground level in the atmosphere are/? and/>0 respectively and v and v0 are the corresponding volumes per unit mass. Given that, if ôp is the infinitesimal increase in pressure due to an increase Oh in height, show that, if the adiabatic relation pv* = p0v0v holds, then Solve this differential equation to find an expression for the pressure at height h.
In calculating both y2(x) and yz(x) we only included terms in the integrand which increased the highest power of / by two over the highest power in the preceding line. The next step gives as required. § 1 : 10] FIRST ORDER DIFFERENTIAL EQUATIONS 55 Example 3. By obtaining an approximate solution of the simultaneous differential equations in terms of powers of x, given that y = 1 and z — \ when x = 0, find the values of y and of z, each correct to four decimal places, when x = | . To apply Picard's method to this problem we must keep the solutions going in step.
8. 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°.
A Course of Mathematics for Engineers and Scientists. Volume 2 by Brian H. Chirgwin and Charles Plumpton (Auth.)