12 taking the laplace transform of the ode we obtain

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Unformatted text preview: can also write . 8. Using partial fractions, . ________________________________________________________________________ page 261 —————————————————————————— CHAPTER 6. —— Hence . 9. The denominator is irreducible over the reals. Completing the square, . Now convert the function to a rational function of the . That is, variable . We find that . Using the fact that , . 10. Note that the denominator the square, function of the variable is irreducible over the reals. Completing . Now convert the function to a rational . That is, . We find that . Using the fact that , . 12. Taking the Laplace transform of the ODE, we obtain . Applying the initial conditions, . Solving for , the transform of the solution is . ________________________________________________________________________ page 262 —————————————————————————— CHAPTER 6. —— Using partial fractions, . Hence . 13. Taking the Laplace transform of the ODE, we obtain . Applying the initial conditions, . Solving for , the transform of the solution is . Since the denominator is irreducible, write the transform as a function of That is, . . First note that . Using the fact that , . Hence . 15. Taking the Laplace transform of the ODE, we obtain . Applying the initial conditions, . Solving for , the transform of the solution is . Since the denominator is irreducible, write the transform as a function of Completing the square, . ________________________________________________________________________ page 263 —————————————————————————— CHAPTER 6. —— . First note that . Using the fact that , the solution of the IVP is . 16. Taking the Laplace transform of the ODE, we obtain . Applying the initial conditions, . Solving for , the transform of the solution is . Since the denominator is irreducible, write the transform as a function of That is, . . We know that . Using the fact that , the solution of the IVP is . 17. Taking the Laplace transform of the ODE, we obtain Applying the initial conditions, ________________________________________________________________________ page 264 —————————————————————————— CHAPTER 6. —— . Solving for the transform of the solution, . Using partial fractions, . Note that of the IVP is and . Hence the solution . 18. Taking the Laplace transform of the ODE, we obtain Applying the initial conditions, . Solving for the transform of the solution, . By inspection, it follows that 19. Taking the Laplace transform of the ODE, we obtain Applying the initial conditions, . Solving for the transform of the solution, . It follows that 20. Taking the Laplace transform of both sides of the ODE, we obtain ________________________________________________________________________ page 265 —————————————————————————— CHAPTER 6. —— . Applying the initial conditions, . Solving for , the transform of the solution is . Using partial fractions on the first term, . First note that and . Hence the solution of the IVP is . 2321. Taking the Laplace transform of both sides of the ODE, we obtain . Applying the initial conditions, . Solving for , the transform of the solution is . Using...
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