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Unformatted text preview: partial fractions on the first term,
.
Thus we can write ________________________________________________________________________
page 266 —————————————————————————— CHAPTER 6. —— .
For the last term, we note that . So that
. We know that
.
Based on the translation property of the Laplace transform,
.
Combining the above, the solution of the IVP is 23. Taking the Laplace transform of both sides of the ODE, we obtain
.
Applying the initial conditions,
.
Solving for , the transform of the solution is
. First write We note that
.
So based on the translation property of the Laplace transform, the solution of the IVP is ________________________________________________________________________
page 267 —————————————————————————— CHAPTER 6. —— 25. Let
be the forcing function on the righthandside. Taking the Laplace
transform
of both sides of the ODE, we obtain
.
Applying the initial conditions,
.
Based on the definition of the Laplace transform, .
Solving for the transform, Using partial fractions, and We find, by inspection, that
.
Referring to Line , in Table ,
. Let
. ________________________________________________________________________
page 268 ————————————————————————— CHAPTER 6. ——
Then . It follows, therefore, that
. Combining the above, the solution of the IVP is
.
26. Let
be the forcing function on the righthandside. Taking the Laplace
transform
of both sides of the ODE, we obtain
.
Applying the initial conditions,
.
Based on the definition of the Laplace transform, .
Solving for the transform, Using partial fractions, We find that
.
Referring to Line , in Table ,
. It follows that ________________________________________________________________________
page 269 —————————————————————————— CHAPTER 6. —— .
Combining the above, the solution of the IVP is
. 28 . Assuming that the conditions of Theorem are satisfied, . Using mathematical induction, suppose that for some ,
. Differentiating both sides, . 29. We know that
.
Based on Prob. ,
. ________________________________________________________________________
page 270 —————————————————————————— CHAPTER 6. ——
Therefore,
. 31. Based on Prob. , Therefore, 33. Using the translation property of the Laplace transform, Therefore, . 34. Using the translation property of the Laplace transform, Therefore, . 35 . Taking the Laplace transform of the given Bessel equation, ________________________________________________________________________
page 271 —————————————————————————— CHAPTER 6. —— .
Using the differentiation property of the transform,
.
That is,
.
It follows that
.
. We obtain a firstorder linear ODE in :
, with integrating factor
.
The firstorder ODE can be written as
,
with solution
.
. In order to obtain negative powers of Expanding , first write in a binomial series,
, valid for . Hence, we can formally express as ________________________________________________________________________
page 272 —————————————————————————— CHAPTER 6...
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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.
 Spring '08
 Staff
 Continuity

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