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The equations reduce to the single equation and a

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Unformatted text preview: ____________________________________ page 409 —————————————————————————— CHAPTER 7. —— I A It can be shown that the partial sums on the left hand side converge for all . Taking the limit as on both sides of the equation, we obtain A Alternatively, consider the system x Ax . Since ODEs are uncoupled, the vectors x e , are a set of linearly independent solutions. Hence the matrix X e e is a fundamental matrix. Finally, since X e e I , it follows that e e 17 . Assuming that x is a solution, then both sides of the equation to obtain A. A , with A x Integrate . Hence x A . . Proceed with the iteration x With A . x , and noting that A is a constant matrix, ________________________________________________________________________ page 410 —————————————————————————— CHAPTER 7. —— x Ax x That is, I Ax A x. . We then have x AI x Ax I A Ax Ax A x. Now suppose that I A A A A A x. It follows that AI AI A A A x A A A A A x x Therefore I A By induction, the asserted form of . Define lim A A x. is valid for all . It can be shown that the...
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