The Existence and Uniqueness Theorem for Linear Systems

The Existence and Uniqueness Theorem for Linear Systems For simplicity, we stick with n = 2, but the results here are true for all n. There are two questions about the following general linear system that we need to consider. � �� � �� � x � = a(t) x + b(t)y x a(t) b(t) x ; in matrix form, = y� = c(t) x + d(t)y y c(t) d(t) y (1) The first is from the previous section: to show that all solutions are of the form x = c1 x1 + x2 x2 , where the xi form a fundamental set, that is, no xi is a constant multiple of the other). (The fact that we can write down all solutions to a linear system in this way is one of the main reasons why such systems are so important.) An even more basic question for the system (1) is: how do we know that it has two linearly independent solutions? For systems with a constant co­ efficient matrix A, we showed in the previous chapters how to solve them explicitly to get two independent solutions. But the general non-constant linear system (1) does not have solutions given by explicit formulas or pro­ cedures. The answers to these questions are based on following theorem. Theorem 2 Existence and uniqueness theorem for linear systems. If the entries of the square matrix A(t) are continuous on an open interval I containing t0 , then the initial value problem x� = A(t) x, x ( t 0 ) = x0 (2) has one and only one solution x(t) on the interval I. The proof is difficult and we shall not attempt it. More important is to see how it is used. The following three theorems answer the questions posed for the 2 × 2 system (1). They are true for n > 2 as well, and the proofs are analogous. In the following theorems, we assume the entries of A(t) are continuous on an open interval I. Here the conclusions are valid on the interval I, fFor example, I could be the whole t-axis. Theorem 2A Linear independence theorem. Let x1 (t) and x2 (t) be two solutions to (1) on the interval I, such that at some point t0 in I, the vectors x1 (t0 ) and x2 (t0 ) are linearly independent. Then a) the solutions x1 (t) and x2 (t) are linearly independent on I, and b) the vectors x1 (t1 ) and x2 (t1 ) are linearly independent at every point t1 of I. Proof. a) By contradiction. If they were dependent on I, one would be a constant multiple of the other, say x2 (t) = c1 x1 (t). Then x2 (t0 ) = c1 x1 (t0 ), � showing them dependent at t0 . b) By contradiction. If there were a point t1 on I where they were de­ pendent, say x2 (t1 ) = c1 x1 (t1 ), then x2 (t) and c1 x1 (t) would be solutions to (1) which agreed at t1 . Hence, by the uniqueness statement in Theorem 2, � x2 (t) = c1 x1 (t) on all of I, showing them linearly dependent on I. Theorem 2B General solution theorem. a) The system (1) has two linearly independent solutions. b) If x1 (t) and x2 (t) are any two linearly independent solutions, then every solution x can be written in the form (3), for some choice of c1 and c2 : (3) x = c 1 x1 + c 2 x2 . Proof. Choose a point t = t0 in the interval I. a) According to Theorem 2, there are two solutions x1 , x2 to (1), satis­ fying respectively the initial conditions x1 (t0 ) = i, x2 (t0 ) = j , (4) where i and j are the usual unit vectors in the xy-plane. Since the two solu­ tions are linearly independent when t = t0 , they are linearly independent on I, by Theorem 5.2A. b) Let u(t) be a solution to (1) on I. Since x1 and x2 are independent at t0 by Theorem 2, using the parallelogram law of addition we can find constants c1� and c2� such that u(t0 ) = c1� x1 (t0 ) + c2� x2 (t0 ). (5) The vector equation (5) shows that the solutions u(t) and c1� x1 (t) + c2� x2 (t) agree at t0 . Therefore by the uniqueness statement in Theorem 2, they are equal on all of I; that is, u(t) = c1� x1 (t) + c2� x2 (t) on I.