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. 8. Using partial fractions,
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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
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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,
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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
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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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 Spring '08
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 Continuity

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