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Assuming that termbyterm inversion is valid, It follows that The series is evidently the expansion, about
36 , of . Taking the Laplace transform of the given Legendre equation,
. Using the differentiation property of the transform,
.
That is, Invoking the initial conditions, we have After carrying out the differentiation, the equation simplifies to That is,
.
37. By definition of the Laplace transform, given the appropriate conditions, ________________________________________________________________________
page 273 —————————————————————————— CHAPTER 6. —— Assuming that the order of integration can be exchanged, Note the region of integration is the area between the lines
Hence and ________________________________________________________________________
page 274 —————————————————————————— CHAPTER 6. ——
Section 6.3
1. 3. 5. ________________________________________________________________________
page 275 —————————————————————————— CHAPTER 6. ——
6. 7.
13 Using the Heaviside function, we can write
. The Laplace transform has the property that
.
Hence
2 .
9. The function can be expressed as Before invoking the of the transform, write the function as It follows that
.
10. It follows directly from the of the transform that
. 11. Before invoking the of the transform, write the function as ________________________________________________________________________
page 276 —————————————————————————— CHAPTER 6. ——
It follows that
.
12. It follows directly from the of the transform that
. 13.
19 Using the fact that ,
. 2115. First consider the function Completing the square in the denominator, It follows that Hence 16. The inverse transform of the function
of the transform, 2217. is Using the First consider the function Completing the square in the denominator, ________________________________________________________________________
page 277 —————————————————————————— CHAPTER 6. —— It follows that Hence 18. Write the function as It follows from the 19 of the transform, that . By definition of the Laplace transform, Making a change of variable, Hence , we have , where . Using the result in Part . , Hence ________________________________________________________________________
page 278 —————————————————————————— CHAPTER 6. ——
. From Part , Note that . Using the fact that ,
. 2620. First write
. Let Based on the results in Prob. ,
, in which 23.
29 . Hence First write Now consider
.
Using the result in Prob. ,
, in which . Hence . It follows that
. 24.
30 By definition of the Laplace transform, ________________________________________________________________________
page 279 —————————————————————————— CHAPTER 6. —— That is, 25.
31 First write the function as . It follows that That is, 26.
32 The transform may be computed directly. On the other hand, using the
of the transform, .
That is,
. 29.
35 The given function is periodic, with . Using the result of Prob. , .
That is, ______________...
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 Spring '08
 Staff
 Continuity

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