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.
 Spring '08
 Staff

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