Unformatted text preview: APTER 6. ——
As , we obtain the formal equation
. Hence
. Setting
in
, and letting
, we find that
These conclusions agree with
the case
, for which it is easy to show that the solution is
.
15 . See Prob. . It follows that the solution of the IVP is
. This function is a multiple of the answer in Prob.
. The maximum value is calculated as
that the appropriate value of is
. Based on Prob. . Hence the peak value occurs at
We find
. , the solution is
. Since this function is a multiple of the solution in Prob.
with
The solution attains a value of
that is,
.
. Similar to Prob. , for , we have
, for ,
, , the solution is
, in which It follows that
. Setting
in
Requiring that the peak value remains at
. These conclusions agree with the case
that the solution is , and letting
, we find that
, the limiting value of is
, for which it is easy to show
. 16 . Taking the initial conditions into consideration, the transformation of the ODE is
. Solving for the transform of the solution, ________________________________________________________________________
page 317 —————————————————————————— CHAPTER 6. —— Using partial fractions, Now let Applying Theorem , the solution is That is, . Consider various values of . For any fixed
If
, then for
, , , as long as It follows that
lim lim Hence
lim . . The Laplace transform of the differential equation
,
with , is
. Solving for the transform of the solution,
________________________________________________________________________
page 318 —————————————————————————— CHAPTER 6. —— It follows that the solution is
.
. 18 . The transform of the ODE given the specified initial conditions is
. Solving for the transform of the solution,
.
Applying Theorem , termbyterm, ________________________________________________________________________
page 319 —————————————————————————— CHAPTER 6. ——
. 19 . Taking the initial conditions into consideration, the transform of the ODE is
. Solving for the transform of the solution,
.
Applying Theorem , termbyterm, . ________________________________________________________________________
page 320 —————————————————————————— CHAPTER 6. —— 20 . The transform of the ODE given the specified initial conditions is
+1 . Solving for the transform of the solution,
+1 Applying Theorem . , termbyterm, . 22 . Taking the initial conditions into consideration, the transform of the ODE is
. Solving for the transform of the solution,
.
Applying Theorem , termbyterm, ________________________________________________________________________
page 321 —————————————————————————— CHAPTER 6. —— . 23 The transform of the ODE given the specified initi...
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 Spring '08
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 Continuity, Laplace, initial conditions

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