X x 8 suppose that the solutions x x x are linearly

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Unformatted text preview: ition of an eigenvector, x x , which implies that is real. 33. From Prob. x Hence we must have , Ax x x Ax . Hence x x x x x x , since the eigenvalues are real. Therefore x Given that , we must have x x x . . ________________________________________________________________________ page 361 —————————————————————————— CHAPTER 7. —— Section 7.4 3. Eq. states that the Wronskian satisfies the first order linear ODE The general solution is , in which is an arbitrary constant. Let X and X be matrices representing two sets of fundamental solutions. It follows that X X Hence X X . Note that 4. First note that As shown in Prob. x , x For second order linear ODE, the Wronskian as defined in Chap. order differential equation . It follows that satisfies the first Alternatively, based on the hypothesis, Direct calculation shows that Here we used the fact that . Hence x 5. The particular solution satisfies the ODE x P x x g Now let ________________________________________________________________________ page 362 ——————————...
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