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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 a1 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 b1 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.
The motion of a certain spring-mass system (see Example 3 of Section 3.8) is described by the second order differential equation
u" + 0.125^ + u = 0. (4)
Rewrite this equation as a system of first order equations.
Let x1 = u and x2 = u'. Then it follows that x[ = x2. Further, u = X2. Then, by substituting for u, u, and u" in Eq. (4), we obtain
x^ + 0.125x2 + x1 = 0.
Thus x1 and x2 satisfy the following system of two first order differential equations:
x1 = x2,
1 2 (5)
x2 = x1 0.125x2.
The general equation of motion of a spring-mass system,
mu + y ti + ku = F (t), (6)
can be transformed to a system of first order equations in the same manner. If we let
x1 = u and x2 = u, and proceed as in Example 1, we quickly obtain the system
xi = x2,
1 2 (7)
= (k / m )x1 (y / m )x2 + F (t)/m.
To transform an arbitrary nth order equation
y(n = F (t, y, y,..., yn~*) (8)
Chapter 7. Systems of First Order Linear Equations
into a system of n first order equations we extend the method of Example 1 by introducing the variables x1, x2,, xn defined by
x1 = y> x2 = y > x3 = y">
It then follows immediately that
xn = y
x2 = x.
xn-1 = xn,
and from Eq. (8)
xn= F(t, x1, x2,..., xn).
Equations (10) and (11) are a special case of the more general system
x1 = F1(t > x1> x2>'--> xn)> x2= F2(t > x1> x2>---> xn)>
xn= Fn (t> x1> x2>---> xn )?
In a similar way the system (1) can be reduced to a system of four first order equations of the form (12), while the systems (2) and (3) are already in this form. In fact, systems of the form (12) include almost all cases of interest, so much of the more advanced theory of differential equations is devoted to such systems.
The system (12) is said to have a solution on the interval I: a < t < i if there exists a set of n functions
Xi = $i(t), X2 =
Xn = (t ),
that are differentiable at all points in the interval I and that satisfy the system of equations (12) at all points in this interval. In addition to the given system of differential equations there may also be given initial conditions of the form
X1(t0) = xl> X2(t0) = X2
Xn (t0) = Xn
where t0 is a specified value of t in I, and x10,, x0 are prescribed numbers. The differential equations (12) and initial conditions (14) together form an initial value problem.
A solution (13) can be viewed as a set of parametric equations in an ^-dimensional space. For a given value of t, Eqs. (13) give values for the coordinates x1,..., xn of a point in the space. As t changes, the coordinates in general also change. The collection of points corresponding to a < t < /3 form a curve in the space. It is often helpful to think of the curve as the trajectory or path of a particle moving in accordance with the system of differential equations (12). The initial conditions (14) determine the starting point of the moving particle.
The following conditions on Fv F2,..., Fn, which are easily checked in specific problems, are sufficient to assure that the initial value problem (12), (14) has a unique solution. Theorem 7.1.1 is analogous to Theorem 2.4.2, the existence and uniqueness theorem for a single first order equation.
X1 = X2
Theorem 7.1.1 Let each of the functions Fv ..., Fn and the partial derivatives d F1/d x1,...,
d F1/dxn,..., d Fn/dxv ..., d Fn/dxn be continuous in a region R of tx1 x2 ??? xn-
space defined by a < t < 3, a1 < x1 < 31,..., an < xn < 3n, and let the point (t0, x°, x°,..., x°) be in R. Then there is an interval |t t0| < h in which there exists a unique solution x1 = 01(t),..., xn = $n(t) of the system of differential equations (12) that also satisfies the initial conditions (14).