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Finally setting i this is precisely the definition of

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Unformatted text preview: ————————— CHAPTER 7. —— Therefore the general solution is . Imposing the initial conditions, we arrive at the equations , with and . Therefore the solution of the IVP is . Since , all solutions converge to the origin. 27 . Suppose that a b the vector , respectively, a 0 . Since a and b are the real and imaginary parts of and b . Hence 0, which leads to 0 Now since and are linearly independent, we must have It follows that . . Recall that u v Consider the equation u a a v b b . 0 , for some . We can then write ________________________________________________________________________ page 400 —————————————————————————— CHAPTER 7. —— a b a b 0. Rearranging the terms, and dividing by the exponential, a From Part b 0. , since a and b are linearly independent, it follows that Without loss of generality, assume that the trigonometric factors are nonzero. Otherwise proceed again from Equation , above. We then conclude that and , which leads to . Thus u and v are linearly independent for some , and...
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