Solving for the transform using partial fractions and

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Unformatted text preview: partial fractions on the first term, . Thus we can write ________________________________________________________________________ page 266 —————————————————————————— CHAPTER 6. —— . For the last term, we note that . So that . We know that . Based on the translation property of the Laplace transform, . Combining the above, the solution of the IVP is 23. Taking the Laplace transform of both sides of the ODE, we obtain . Applying the initial conditions, . Solving for , the transform of the solution is . First write We note that . So based on the translation property of the Laplace transform, the solution of the IVP is ________________________________________________________________________ page 267 —————————————————————————— CHAPTER 6. —— 25. Let be the forcing function on the right-hand-side. Taking the Laplace transform of both sides of the ODE, we obtain . Applying the initial conditions, . Based on the definition of the Laplace transform, . Solving for the transform, Using partial fractions, and We find, by inspection, that . Referring to Line , in Table , . Let . ________________________________________________________________________ page 268 ————————————————————————— CHAPTER 6. —— Then . It follows, therefore, that . Combining the above, the solution of the IVP is . 26. Let be the forcing function on the right-hand-side. Taking the Laplace transform of both sides of the ODE, we obtain . Applying the initial conditions, . Based on the definition of the Laplace transform, . Solving for the transform, Using partial fractions, We find that . Referring to Line , in Table , . It follows that ________________________________________________________________________ page 269 —————————————————————————— CHAPTER 6. —— . Combining the above, the solution of the IVP is . 28 . Assuming that the conditions of Theorem are satisfied, . Using mathematical induction, suppose that for some , . Differentiating both sides, . 29. We know that . Based on Prob. , . ________________________________________________________________________ page 270 —————————————————————————— CHAPTER 6. —— Therefore, . 31. Based on Prob. , Therefore, 33. Using the translation property of the Laplace transform, Therefore, . 34. Using the translation property of the Laplace transform, Therefore, . 35 . Taking the Laplace transform of the given Bessel equation, ________________________________________________________________________ page 271 —————————————————————————— CHAPTER 6. —— . Using the differentiation property of the transform, . That is, . It follows that . . We obtain a first-order linear ODE in : , with integrating factor . The first-order ODE can be written as , with solution . . In order to obtain negative powers of Expanding , first write in a binomial series, , valid for . Hence, we can formally express as ________________________________________________________________________ page 272 —————————————————————————— CHAPTER 6...
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This note was uploaded on 03/11/2014 for the course MA 303 taught by Professor Staff during the Spring '08 term at Purdue.

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