Unformatted text preview: ________________________________________________________________________
page 179 —————————————————————————— CHAPTER 5. —— .
Setting the coefficients equal to zero, we obtain , , and ,
Observe that for
two, we also have and
for . , we obtain
. Since the indices differ by
. Therefore the general solution is a polynomial
. Hence the linearly independent solutions are
and
10. Let .
. Then Substitution into the ODE results in First write
.
It follows that
.
We obtain , and
, . Note that for
,
. Since the indices differ by two, we also have
. On the other hand, for
, for Therefore the general solution is
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page 180 —————————————————————————— CHAPTER 5. —— .
Hence the linearly independent solutions are and
. 11. Let . Then and Substitution into the ODE results in
.
Before proceeding, write and It follows that
.
We obtain , , and
, . The indices differ by two, so for and
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page 181 —————————————————————————— CHAPTER 5. —— Hence the linearly independent solutions are 12. Let . Then and Substitution into the ODE results in
.
Before proceeding, write and It follows that
.
We obtain for and . Writing out the individual equations, ________________________________________________________________________
page 182 —————————————————————————— CHAPTER 5. —— The coefficients can be calculated successively as
,
,
, . We can now see that for
,
is
proportional to . In fact, for
,
. Therefore the general solution is
.
Hence the linearly independent solutions are 13. Let and . Then and Substitution into the ODE results in
.
First write We then obtain It follows that and ________________________________________________________________________
page 183 —————————————————————————— CHAPTER 5. ——
for . The indices differ by two, so for and Hence the linearly independent solutions are 15 . From Prob. , we have
and Since and , we have .
. That is,
. The four and fiveterm polynomial approximations are ________________________________________________________________________
page 184 —————————————————————————— CHAPTER 5. ——
. . The fourterm approximation
on the interval
.
17 Since appears to be reasonably accurate within % . From Prob. , the linearly independent solutions are and , we have . That is,
. The four and fiveterm polynomial approximations are ________________________________________________________________________
page 185 —————————————————————————— CHAPTER 5. ——
. . The fourterm approximation
on the interval
.
18 . From Prob. appears to be reasonably accurate within , we have
and Since an...
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 Spring '08
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 Taylor Series, lim, Complex number

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