Solving for the transform of the solution

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Unformatted text preview: g ceases, 6. Taking the Laplace transform of both sides of the ODE, we obtain . Applying the initial conditions, . Solving for the transform, . Using partial fractions, ________________________________________________________________________ page 288 —————————————————————————— CHAPTER 6. —— and Taking the inverse transform. term-by-term, the solution of the IVP is Due to the initial conditions, the response has a transient overshoot, followed by an exponential convergence to a steady value of . 7. Taking the Laplace transform of both sides of the ODE, we obtain . Applying the initial conditions, ________________________________________________________________________ page 289 —————————————————————————— CHAPTER 6. —— . Solving for the transform, . Using partial fractions, . Hence . Taking the inverse transform, the solution of the IVP is Due to initial conditions, the solution temporarily oscillates about forcing is applied, the response is a steady oscillation about . . After the ________________________________________________________________________ page 290 —————————————————————————— CHAPTER 6. —— 9. 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, . The forcing function can be written as with Laplace transform . Solving for the transform, . Using partial fractions, Taking the inverse transform, and using Theorem , the solution of the IVP is ________________________________________________________________________ page 291 —————————————————————————— CHAPTER 6. —— The solution increases, in response to the ramp input, and thereafter oscillates about a mean value of . 11. Taking the Laplace transform of both sides of the ODE, we obtain . Applying the initial conditions, . Solving for the transform, . Using partial fractions, Taking the inverse transform, and applying Theorem , . ________________________________________________________________________ page 292 —————————————————————————— CHAPTER 6. —— Since there is no damping term, the solution responds immediately to the forcing input. There is a temporary oscillation about 12. Taking the Laplace transform of the ODE, we obtain Applying the initial conditions, . Solving for the transform of the solution, . Using partial fractions, It follows that ________________________________________________________________________ page 293 —————————————————————————— CHAPTER 6. —— Based on Theorem , the solution of the IVP is . The solution increases without bound, exponentially. 13. Taking the Laplace transform of the ODE, we obtain Applying the initial conditions, . Solving for the transform of the solution, ________________________________________________________________________ page 294 —————————————————————————— CHAPTER 6. —— . Using partial fractions, It follows that Based on Theorem , the solution of the IVP is . That is, . ________________________________________________________________________ page 295 —————————————————————————...
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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 University-West Lafayette.

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