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X x x 0 the collection of vectors x is linearly

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Unformatted text preview: ———————————————— CHAPTER 7. —— x be any solution of the homogeneous equation. That is, x P x . We know that x x , in which x is a linear combination of some fundamental solution. By linearity of the differential equation, it follows that x x x is a solution of the ODE. Based on the uniqueness theorem, all solutions must have this form. 7 . By definition, x x . The Wronskian vanishes at independent on . It follows from Theorem must be discontinuous at and . Hence the vectors are linearly that one or more of the coefficients of the ODE and . If not, the Wronskian would not vanish. . Let x Then x On the other hand, x Comparing coefficients, we find that Solution of this system of equations results in Hence the vectors are solutions of the ODE ________________________________________________________________________ page 363 —————————————————————————— CHAPTER 7. —— x x 8. Suppose that the solutions x , x , , x are linearly dependent at there are constants not all...
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