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Unformatted text preview: hat 24. Clearly, ________________________________________________________________________
page 172 —————————————————————————— CHAPTER 5. —— Shifting the index in the first series, that is, setting , Hence
.
Note that when and , the coefficients in the second series are zero. So that 26. Clearly, Shifting the index in the first series, that is, setting , Shifting the index in the second series, that is, setting Combining the series, and starting the summation at , , 27. We note that ________________________________________________________________________
page 173 —————————————————————————— CHAPTER 5. —— Shifting the index in the first series, that is, setting , ,
since the coefficient of the term associated with is zero. Combining the series, ________________________________________________________________________
page 174 —————————————————————————— CHAPTER 5. ——
Section 5.2
1. Let . Then Substitution into the ODE results in or Equating all the coefficients to zero,
,
We obtain the recurrence relation
,
The subscripts differ by two, so for and Hence The linearly independent solutions are . ________________________________________________________________________
page 175 —————————————————————————— CHAPTER 5. ——
4. Let . Then Substitution into the ODE results in
.
Rewriting the second summation,
,
that is, Setting the coefficients equal to zero, we have ,
, , and for The recurrence relation can be written as
,
The indices differ by four, so , , ,
Similarly, , , , , . are defined by
, , . , , . are defined by
, The remaining coefficients are zero. Therefore the general solution is .
Note that for the even coefficients, ________________________________________________________________________
page 176 —————————————————————————— CHAPTER 5. —— ,
and for the odd coefficients,
, . Hence the linearly independent solutions are 6. Let . Then and Substitution into the ODE results in
.
Before proceeding, write and It follows that
. ________________________________________________________________________
page 177 —————————————————————————— CHAPTER 5. ——
Equating the coefficients to zero, we find that ,
, , and
. The indices differ by two, so for and Hence the linearly independent solutions are 8.
7 Let . Then and Substitution into the ODE results in
.
First write We then obtain ________________________________________________________________________
page 178 —————————————————————————— CHAPTER 5. ——
It follows that
and
indices differ by two, so for , . Note that the and Hence the linearly independent solutions are 9. Let . Then and Substitution into the ODE results in
.
Before proceeding, write and It follows that...
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 Spring '08
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