Furthermore also converges absolutely for all term by

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Unformatted text preview: ind that . In order to solve the second equation, set remaining equations results in , , Hence a second solution is . Solution of the ,. 16 . After multiplying both sides of the ODE by , we find that . Both of these functions are analytic at , hence singular point. . Furthermore, and . . So the indicial equation is . In order to find the solution corresponding to and is a regular , with roots , set Upon substitution into the ODE, we have . That is, . Setting the coefficients equal to zero, we find that for , . It follows that ________________________________________________________________________ page 239 —————————————————————————— CHAPTER 5. —— . Hence one solution is . The exponents differ by an integer. So for a second solution, set . Substituting into the ODE, we obtain . Since , it follows that . Now Substituting for , the right hand side of the ODE is . Equating the coefficients, we obtain the system of equations Evidently, that . In order to solve the second equation, set . We then find , , , . Therefore a second solution is ________________________________________________________________________ page 240 —————————————————————————— CHAPTER 5. —— 19 . After dividing by the leading coefficient, we find that lim lim lim Hence with roots For lim is a regular singular point. The indicial equation is and . , lim lim lim Hence , lim is a regular singular point. The indicial equation is , with roots . Given that and . is not a positive integer, we can set Substitution into the ODE results in That is, Combining the series, we obtain , in which ________________________________________________________________________ page 241 —————————————————————————— CHAPTER 5. —— . Note that equal to zero, we have for . Setting the coefficients , and . Hence one solution is Since the nearest other singularity is at be at least . . Given that , the radius of convergence of is not a positive integer, we can set will Then Substitution into the ODE results in That is, After adjusting the indices, Combining the series, we obtain , in which ________________________________________________________________________ page 242 —————————————————————————— CHAPTER 5. —— . Note that it follows that for . Setting , , Therefore a second solution is . Under the transformation , the ODE becomes . That is, . Therefore is a singular point. Note that and . It follows that lim lim lim Hence , lim is a regular singular point. The indicial equation is , or 21 Evidently, the roots are and . . Note that and . ________________________________________________________________________ page 243 —————————————————————————— CHAPTER 5. —— I...
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