Download (direct link):
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 Anafysis 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 xv 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 Fl(t) and F2(t), and they are also constrained by the three springs whose constants are kj, 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:
ò1^Ò = k2(x2 - Xl) - k1 X1 + F1(t)
= -(k1 + k2)x1 + k2 x2 + F1 (t),
= -k3x2 - k2(x2 - x1) + 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
FIGURE 7.1.2 A parallel LRC circuit.
the prey will increase. A mathematical model showing how an ecological balance can be maintained when both are present was proposed about 1925 by Lotka and Volterra. The model consists of the system of differential equations
dH/ dt = a, H — b, HP,
1 1 (3)
dP/dt = — a2 P + b2 HP,
known as the predator-prey equations. In Eqs. (3) the coefficient ax is the birth rate of the population H; similarly, a2 is the death rate of the population P. The HP terms in the two equations model the interaction of the two populations. The number of encounters between predator and prey is assumed to be proportional to the product of the populations. Since any such encounter tends to be good for the predator and bad for the prey, the sign of the HP term is negative in the first equation and positive in the second. The coefficients bx and b2 are the coefficients of interaction between predator and prey.
Another reason for the importance of systems of first order equations is that equations of higher order can always be transformed into such systems. This is usually required if a numerical approach is planned, because almost all codes for generating approximate numerical solutions of differential equations are written for systems of first order equations. The following example illustrates how easy it is to make the transformation.