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. termbyterm, 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 righthandside. 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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 Spring '08
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 Continuity, Laplace, initial conditions

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