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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' + 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
X2 + 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" + ó u + 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
*! = x2,
1 2 (7)
x2 = —(k / m )xj — (y / m )x2 + F (t)/m.
To transform an arbitrary nth order equation
7(n) = F (t, 7, 7,..., 7(n—1)) (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 = / > x3 = /’
It then follows immediately that
xn = y
4-1 = 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: à < t < â if there exists a set of n functions
xx = Ô(), x2 = ô2(³),
xn = Ôï (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 x0,..., x° 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 n-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 à < t < â 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 F1, 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/äx1,..., d Fn/äxn be continuous in a region R of tx1x2 ¦ ¦¦ xn-
space defined by a < t < â, a1 < x1 < â1, ... ,an < xn < âï, 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 = ô1^),..., xn = Ôï (t) of the system of differential equations (12) that also satisfies the initial conditions (14).
The proof of this theorem can be constructed by generalizing the argument in Section 2.8, but we do not give it here. However, note that in the hypotheses of the theorem nothing is said about the partial derivatives of F1,..., Fn with respect to the independent variable t. Also, in the conclusion, the length 2h of the interval in which the solution exists is not specified exactly, and in some cases may be very short. Finally, the same result can be established on the basis of somewhat weaker, but more complicated hypotheses, so the theorem as stated is not the most general one known, and the given conditions are sufficient, but not necessary, for the conclusion to hold.