Unformatted text preview: is a regular singular point. The indicial equation is given by
,
that is, , with roots . Let .
. Substitution into the ODE results in
. After adjusting the indices in the first series, we obtain
.
Setting the coefficients equal to zero, it follows that for , .
So for , ________________________________________________________________________
page 234 —————————————————————————— CHAPTER 5. —— With , one solution is
. For a second solution, set
Substituting into the ODE, we obtain . .
Since , it follows that
. More specifically, .
Equating the coefficients, we obtain the system of equations Solving these equations for the coefficients,
,
, . Therefore a second solution is 14 . Here
, therefore , , and
Both of these functions are analytic at
is a regular singular point. Note that The indicial equation is given by ________________________________________________________________________
page 235 —————————————————————————— CHAPTER 5. ——
,
that is, , with roots and . In order to find the solution corresponding to .
, set Upon substitution into the ODE, we have
.
After adjusting the indices in the first two series, and expanding the exponential function, .
Equating the coefficients, we obtain the system of equations Setting
, , solution of the system results in
. Therefore 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
. More specifically, ________________________________________________________________________
page 236 —————————————————————————— CHAPTER 5. —— .
Equating the coefficients, we obtain the system of equations Solving these equations for the coefficients,
equations, set
. Then
,
Therefore a second solution is 15 . Note the . In order to solve the remaining
,
, and . Furthermore,
It follows that and
lim
lim
and therefore is a regular singular point. The indicial equation is given by
,
that is, , with roots and . In order to find the solution corresponding to .
, set Upon substitution into the ODE, we have ________________________________________________________________________
page 237 —————————————————————————— CHAPTER 5. —— .
After adjusting the indices, it follows that .
That is, Setting the coefficients equal to zero, we have , and for ,
. If we assign
, then we obtain
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 ________________________________________________________________________
page 238 —————————————————————————— CHAPTER 5. —— .
Equating the coefficients, we obtain the system of equations We f...
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 Spring '08
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 Taylor Series, lim, Complex number

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