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Recall from Section 7.3 that the columns of X(t) are linearly independent for a given value of t if and only if det X = 0 for that value of t. This determinant is called the Wronskian of the n solutions x(1),..., x(n) and is also denoted by W[x(1), that is,
](t) = det X(t).
The solutions x(1), W[ x
x(n)] is not zero there.
are then linearly independent at a point if and only if
If the vector functions x(1),..., x(n) are linearly independent solutions of the system (3) for each point in the interval a < t < โ, then each solution x = ๔(^ of the system (3) can be expressed as a linear combination of x(1),..., x(n),
๔(0 = c1x(1)(t) + + cn x(n)(t), (11)
in exactly one way.
Chapter 7. Systems ofFirst Order Linear Equations
Before proving Theorem 7.4.2, note that according to Theorem 7.4.1 all expressions of the form (11) are solutions of the system (3), while by Theorem 7.4.2 all solutions of Eq. (3) can be written in the form (11). If the constants cv , cn are thought of as arbitrary, then Eq. (11) includes all solutions of the system (3), and it is customary to call it the general solution. Any set of solutions x(1),..., x(n) of Eq. (3), which is linearly independent at each point in the interval a < t < โ, is said to be a fundamental set of solutions for that interval.
To prove Theorem 7.4.2, we will show, given any solution ๔ of Eq. (3), that ๔(0 = cxx(1)(t) + ฆ + cnx(n)(t) for suitable values of cv ..., cn. Let t = t0 be some point
in the interval a < t < โ and let g = ๔^). We now wish to determine whether there is any solution of the form x = cxx(1)(t) + ฆ + cnx(n)(t) that also satisfies the same
initial condition x(t0) = g. That is, we wish to know whether there are values of c1t..., cn such that
c1x(1)(t0) + ... + cn x(n)(t0) = g, (12)
or in scalar form
c1 x11(t0) + + cnx1n (t0) = %\
c1 xnl(t0) + " + cnxnn(t0) = %n.
The necessary and sufficient condition that Eqs. (13) possess a unique solution ct,..., cn is precisely the nonvanishing of the determinant of coefficients, which is the Wronskian W[ x(1),..., x(n)] evaluated at t = t0. The hypothesis that x(1),..., x(n) are linearly independent throughout a < t < โ guarantees that W[x(1),..., x(n)] is not zero at t = t0, and therefore there is a (unique) solution of Eq. (3) of the
form x = cxx(1)(t) + + cnx(n)(t) that also satisfies the initial condition (12). By
the uniqueness part of Theorem 7.1.2 this solution is identical to ๔(0, and hence ๔^) = cl x(1)(t) + + cnx(n)(t), as was to be proved.
If x(1),..., x(n) are solutions of Eq. (3) on the interval a < t < โ, then in this interval W[x(1),..., x(n)] either is identically zero or else never vanishes.
The significance of Theorem 7.4.3 lies in the fact that it relieves us of the necessity of examining W[x(1),..., x(n)] at all points in the interval of interest, and enables us to determine whether x(1),..., x(n) form a fundamental set of solutions merely by evaluating their Wronskian at any convenient point in the interval.
Theorem 7.4.3 is proved by first establishing that the Wronskian of x(1),..., x(n) satisfies the differential equation (see Problem 2)
= (pn + P22 + + Pnn) W. (14)
Hence W is an exponential function, and the conclusion of the theorem follows immediately. The expression for W obtained by solving Eq. (14) is known as Abels formula; note the analogy with Eq. (8) of Section 3.3.
Alternatively, Theorem 7.4.3 can be established by showing that if n solutions x(1),..., x(n) of Eq. (3) are linearly dependent at one point t = t0, then they must
7.4 Basic Theory of Systems of First Order Linear Equations
be linearly dependent at each point in เ < t < โ (see Problem 8). Consequently, if x(1),..., x(n) are linearly independent at one point, they must be linearly independent at each point in the interval.
The next theorem states that the system (3) always has at least one fundamental set of solutions.
To prove this theorem, note that the existence and uniqueness of the solutions x(1),..., x(n) mentioned in Theorem 7.4.4 are assured by Theorem 7.1.2. It is not hard to see that the Wronskian of these solutions is equal to 1 when t = t0; therefore x(1),..., x(n) are a fundamental set of solutions.
Once one fundamental set of solutions has been found, other sets can be generated by forming (independent) linear combinations of the first set. For theoretical purposes the set given by Theorem 7.4.4 is usually the simplest.