So that 26 clearly shifting the index in the first

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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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