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The geometrical configuration is shown in Figure 6.6.2. The starting point P(a, b) is joined to the terminal point (0, 0) by the arc C. Arc length s is measured from the origin, and f (y) denotes the rate of change of s with respect to y:
Then it follows from the principle of conservation of energy that the time T(b) required for a particle to slide from P to the origin is
(a) Show that if u is a function such that u"(t) — 0(t), then
u"(t) + u(t) — tu'(o) — u(o) — sin2t.
u" (t) + u(t) — sin2t; u(o) — o, u'(o) — o.
6.6 The Convolution Integral
(a) Assume that T(b) = T0, a constant, for each b. By taking the Laplace transform of Eq. (ii) in this case and using the convolution theorem, show that
ft — .
then show that
f (y) = /-—/
2g —L n —y'
Hint: See Problem 27 of Section 6.1.
(b) Combining Eqs. (i) and (iv), show that
2a — y
where a = gT^/n2.
(c) Use the substitution y = 2a sin2 (9/2) to solve Eq. (v), and show that x = a(9 + sin 9), y = a(1 — cos 9).
Equations (vi) can be identified as parametric equations of a cycloid. Thus the tautochrone is an arc of a cycloid.
The books listed below contain additional information on the Laplace transform and its applications: Churchill, R. V., Operational Mathematics (3rd ed.) (New York: McGraw-Hill, 1971).
Doetsch, G., Nader, W. (tr.), Introduction to the Theory and Application of the Laplace Transform (New York: Springer-Verlag, 1974).
Kaplan, W., Operational Methods for Linear Systems (Reading, MA: Addison-Wesley, 1962). Kuhfittig, P. K. F., Introduction to the Laplace Transform (New York: Plenum, 1978).
Miles, J. W., Integral Transforms in Applied Mathematics (London: Cambridge University Press, 1971).
Rainville, E. D., The Laplace Transform: An Introduction (New York: Macmillan, 1963).
Each of the books just mentioned contains a table of transforms. Extensive tables are also available; see, for example:
Erdelyi, A. (ed.), Tables of Integral Transforms (Vol. 1) (New York: McGraw-Hill, 1954).
Chapter 6. The Laplace Transform
Roberts, G. E., and Kaufman, H., Table of Laplace Transforms (Philadelphia: Saunders, 1966).
A further discussion of generalized functions can be found in:
Lighthill, M. J., Fourier Analysis and Generalized Functions (London: Cambridge University Press, 1958).
Systems of First Order Linear Equations
There are many physical problems that involve a number of separate elements linked together in some manner. For example, electrical networks have this character, as do some problems in mechanics or in other fields. In these and similar cases the corresponding mathematical problem consists of a system of two or more differential equations, which can always be written as first order equations. In this chapter we focus on systems of first order linear equations, utilizing some of the elementary aspects of linear algebra to unify the presentation.
Systems of simultaneous ordinary differential equations arise naturally in problems involving several dependent variables, each of which is a function of a single independent variable. We will denote the independent variable by t, and let x1, x2, x3,... represent dependent variables that are functions of t. Differentiation with respect to t will be denoted by a prime.
For example, consider the spring-mass system in Figure 7.1.1. The two masses move on a frictionless surface under the influence of external forces F1(t) and F2(t), and they are also constrained by the three springs whose constants are k1, k2, and k3,
Chapter 7. Systems of First Order Linear Equations
respectively. Using arguments similar to those in Section 3.8, we find the following equations for the coordinates x1 and x2 of the two masses:
^1^2" = k2(X2 - Xl) - k1 X1 + F1(t)
= -(k1 + k2)x1 + k2 x2 + F1 (t ),
= -k3x2 - k2(x2 - X") + F2(t) = k2x1 - (k2 + k3)x2 + F2(t).
A derivation of Eqs. (1) is outlined in Problem 17.
FIGURE 7.1.1 A two degrees of freedom spring-mass system.
Next, consider the parallel LRC circuit shown in Figure 7.1.2. Let V be the voltage drop across the capacitor and I the current through the inductor. Then, referring to Section 3.8 and to Problem 18 of this section, we can show that the voltage and current are described by the system of equations
dt L dV
where L is the inductance, C the capacitance, and R the resistance.
As a final example, we mention the predator-prey problem, one of the fundamental problems of mathematical ecology, which is discussed in more detail in Section 9.5. Let H(t) and P(t) denote the populations at time t of two species, one of which (P) preys on the other (H). For example, P(t) and H(t) may be the number of foxes and rabbits, respectively, in a woods, or the number of bass and redear (which are eaten by bass) in a pond. Without the prey the predators will decrease, and without the predator