For lim lim is a regular singular point for lim lim

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Unformatted text preview: ation . Multiplying both sides by and integrating, . Based on Prob. , . Hence ________________________________________________________________________ page 203 —————————————————————————— CHAPTER 5. —— Section 5.4 2. 18 We see that when and . Since the three coefficients have no factors in common, both of these points are singular points. Near , lim lim The singular point lim lim is regular. Considering lim , lim The latter limit does not exist. Hence is an irregular singular point. 3 20. when and . Since the three coefficients have no common factors, both of these points are singular points. Near , lim The limit does not exist, and so lim is an irregular singular point. Considering lim lim lim Hence , lim is a regular singular point. 4. 19 when and . Since the three coefficients have no common factors, both of these points are singular points. Near , lim The limit does not exist, and so lim lim is an irregular singular point. Near , lim ________________________________________________________________________ page 204 —————————————————————————— CHAPTER 5. —— lim Hence lim is a regular singular point. At lim lim lim Hence 6. 22 lim is a regular singular point. The only singular point is at . We find that lim lim lim Hence 7. 25 lim is a regular singular point. The only singular point is at . We find that lim lim lim Hence 8. 24 , lim is a regular singular point. Dividing the ODE by , we find that and The singular points are at and lim lim Hence . For . , lim lim is a regular singular point. For , ________________________________________________________________________ page 205 —————————————————————————— CHAPTER 5. —— lim lim lim Hence lim is a regular singular point. For lim lim lim lim The latter limit does not exist. Hence 9 23 . , is an irregular singular point. Dividing the ODE by , we find that and The singular points are at and . For lim , lim The limit does not exist. Hence is an irregular singular point. For lim , lim lim Hence . lim is a regular singular point. 10. 26 when and . Since the three coefficients have no common factors, both of these points are singular points. Near , lim lim Hence lim lim is a regular singular point. For , ________________________________________________________________________ page 206 —————————————————————————— CHAPTER 5. —— lim lim lim Hence 11. 27 lim is a regular singular point. Dividing the ODE by , we find that and The singular points are at and . For lim lim is a regular singular point. For lim , lim lim Hence , lim lim Hence . lim is a regular singular point. 29 13. Note that and . Evidently, is not analytic at Furthermore, the function does not have a Taylor series about Hence is an irregular singular point. . . 30 14. when . Since the three coefficients have no common factors, is a singular point. The Taylor series of , about , is . Hence the function...
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This note was uploaded on 03/11/2014 for the course MA 303 taught by Professor Staff during the Spring '08 term at Purdue.

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