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Unformatted text preview: series of
, about is analytic at . Similarly, the Taylor , is
. The function
singular point. is also analytic at . Hence is a regular ________________________________________________________________________
page 207 —————————————————————————— CHAPTER 5. ——
15.
32 when
. Since the three coefficients have no common factors,
is a singular point. The Taylor series of
, about
, is
.
Hence the function
is a rational function, with is analytic at lim
Hence . On the other hand, lim is a regular singular point. 16.
31 when
. Since the three coefficients have no common factors,
is a singular point. We find that
lim lim Although the function does not have a Taylor series about
, note that
. Hence
is a regular
singular point. Furthermore,
is undefined at
. Therefore the
points
are also singular points. First note that
lim
Furthermore, since has period lim
, Therefore From above,
.
Note that the function in brackets is analytic near
. It follows that the function
is also analytic near
. Hence all the singular points are regular. 318.
4 The singular points are located at
,
. Dividing the ODE by
, we find that
and
. Evidently,
is
not even defined at
. Hence
is an irregular singular point. On the other
hand, the Taylor series of
, about
, is ________________________________________________________________________
page 208 —————————————————————————— CHAPTER 5. —— .
Noting that , It is apparent that is analytic at . Similarly,
, which is also analytic at 42
20.
and . Hence all other singular points are regular. is the only singular point. Dividing the ODE by
It follows that
lim lim lim , we have , lim Hence
is a regular singular point. Let
Substitution into the ODE results in . .
That is,
.
It follows that Equating the coefficients to zero, we find that ,
, We conclude that all the
can be obtained.
22. Based on Prob. are equal to zero. Hence , the change of variable, , and
.
is the only solution that
, transforms the ODE into the ________________________________________________________________________
page 209 —————————————————————————— CHAPTER 5. ——
form
.
Evidently,
of lim 424.
8 is a singular point. Now
does not exist,
, that is, Under the transformation and
. Since the value
, is an irregular singular point. , the ODE becomes
, that is,
.
Therefore is a singular point. Note that
and . It follows that
lim lim
Hence
26.
46 lim , lim is a regular singular point. Under the transformation , the ODE becomes
, that is,
.
Therefore is a singular point. Note that
and It immediately follows that the limit lim . does not exist. Hence ________________________________________________________________________
page 210 —————————————————————————— CHAPTER 5. ——
is an irregular singular point.
27.
49 Under the transformation , the ODE becomes
. Therefore is a singular point. Note that
and . We find that
lim lim , but
lim
The latter limit does not exist. Hence lim .
is an irregul...
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 Spring '08
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