The indicial equation is given by that is with roots

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Unformatted text preview: into the ODE results in After adjusting the indices in the second-to-last series, we obtain Assuming , the indicial equation is , with roots and . Setting the remaining coefficients equal to zero, the recurrence relation is , . Setting the remaining coefficients equal to zero, we have of the odd coefficients are zero. With , , So for , which implies that all . , ________________________________________________________________________ page 225 —————————————————————————— CHAPTER 5. —— . With , , So for . , . The two linearly independent solutions are 9. Note that Set and , which are both analytic at . . 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 and . Setting the remaining coefficients equal to zero, the recurrence relation is , With . , ________________________________________________________________________ page 226 —————————————————————————— CHAPTER 5. —— , It follows that for . , . Therefore one solution is . 10. Here Set and , which are both analytic at . . Substitution into the ODE results in After adjusting the indices in the second series, we obtain Assuming , the indicial equation is , with roots Setting the remaining coefficients equal to zero, we find that . The recurrence relation is , With . , , Since , the odd coefficients are zero. So for . , . Therefore one solution is ________________________________________________________________________ page 227 —————————————————————————— CHAPTER 5. —— . 12 . Dividing through by the leading coefficient, the ODE can be written as . For , lim lim For lim lim , lim lim lim lim Hence both and are regular singular points. As shown in Example , the indicial equation is given by . In this case, both sets of roots are . Let , and equation becomes Based on Part , and . Under this change of variable, the differential is a regular singular point. Set Substitution into the ODE results in Upon inspection, we can also write ________________________________________________________________________ page 228 —————————————————————————— CHAPTER 5. —— After adjusting the indices in the second series, it follows that Assuming that , the indicial equation is The recurrence relation is , with roots , . , With With , we find that for , we find that for . , , The two linearly independent solutions of the Chebyshev equation are 13. Here In fact, and lim Hence the indicial equation is , which are both analytic at and lim , with roots . . . Set ________________________________________________________________________ page 229 —————————————————————————— CHAPTER 5. —— . Substitution into the ODE results in That is, It follows that . Setting the coef...
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