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Solving for the transform of the solution as shown in

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Unformatted text preview: APTER 6. —— As , we obtain the formal equation . Hence . Setting in , and letting , we find that These conclusions agree with the case , for which it is easy to show that the solution is . 15 . See Prob. . It follows that the solution of the IVP is . This function is a multiple of the answer in Prob. . The maximum value is calculated as that the appropriate value of is . Based on Prob. . Hence the peak value occurs at We find . , the solution is . Since this function is a multiple of the solution in Prob. with The solution attains a value of that is, . . Similar to Prob. , for , we have , for , , , the solution is , in which It follows that . Setting in Requiring that the peak value remains at . These conclusions agree with the case that the solution is , and letting , we find that , the limiting value of is , for which it is easy to show . 16 . Taking the initial conditions into consideration, the transformation of the ODE is . Solving for the transform of the solution, ________________________________________________________________________ page 317 —————————————————————————— CHAPTER 6. —— Using partial fractions, Now let Applying Theorem , the solution is That is, . Consider various values of . For any fixed If , then for , , , as long as It follows that lim lim Hence lim . . The Laplace transform of the differential equation , with , is . Solving for the transform of the solution, ________________________________________________________________________ page 318 —————————————————————————— CHAPTER 6. —— It follows that the solution is . . 18 . The transform of the ODE given the specified initial conditions is . Solving for the transform of the solution, . Applying Theorem , term-by-term, ________________________________________________________________________ page 319 —————————————————————————— CHAPTER 6. —— . 19 . Taking the initial conditions into consideration, the transform of the ODE is . Solving for the transform of the solution, . Applying Theorem , term-by-term, . ________________________________________________________________________ page 320 —————————————————————————— CHAPTER 6. —— 20 . The transform of the ODE given the specified initial conditions is +1 . Solving for the transform of the solution, +1 Applying Theorem . , term-by-term, . 22 . Taking the initial conditions into consideration, the transform of the ODE is . Solving for the transform of the solution, . Applying Theorem , term-by-term, ________________________________________________________________________ page 321 —————————————————————————— CHAPTER 6. —— . 23 The transform of the ODE given the specified initi...
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