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# Setting the remaining coefficients equal to zero the

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Unformatted text preview: icients have no common factors, is a singular point. Near , lim lim lim Hence lim is a regular singular point. Let . Then and Substitution into the ODE results in . That is, . It follows that Assuming that , we obtain the indicial equation , with roots ________________________________________________________________________ page 220 —————————————————————————— CHAPTER 5. —— and . It immediately follows that equal to zero, we have . Setting the remaining coefficients , For . , the recurrence relation becomes , Since For . , the odd coefficients are zero. Furthermore, for , , the recurrence relation becomes , Since . , the odd coefficients are zero, and for , The two linearly independent solutions are 3 5. Note that and , which are both analytic at . Set . Substitution into the ODE results in , and after multiplying both sides of the equation by , . It follows that ________________________________________________________________________ page 221 —————————————————————————— CHAPTER 5. —— Setting the coefficients equal to zero, the indicial equation is are and . Here . The recurrence relation is , For . The roots . , , Hence for . , . Therefore one solution is . 5. 3 Here and , which are both analytic at . Set . Substitution into the ODE results in It follows that . Assuming , the indicial equation is , with roots Setting the remaining coefficients equal to zero, we have , and , , . It immediately follows that the odd coefficients are equal to zero. For , So for , . , ________________________________________________________________________ page 222 —————————————————————————— CHAPTER 5. —— . For , , So for . , . The two linearly independent solutions are 6. Note that and , which are both analytic at . Set . Substitution into the ODE results in After adjusting the indices in the second-to-last series, we obtain Assuming , the indicial equation is , with roots the remaining coefficients equal to zero, the recurrence relation is , . Setting . First note that . So for , , . ________________________________________________________________________ page 223 —————————————————————————— CHAPTER 5. —— It follows that , For . , , , and therefore , . The two linearly independent solutions are 7. Here and After multiplying both sides by , which are both analytic at . Set . Substitution into the ODE results in , +1 +1 After adjusting the indices in the last two series, we obtain ________________________________________________________________________ page 224 —————————————————————————— CHAPTER 5. —— Assuming , the indicial equation is , with roots the remaining coefficients equal to zero, the recurrence relation is , . , With . Setting . , Hence one solution is . 8. Note that Set and , which are both analytic at . . Substitution...
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