This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ”â€”â€”â€”â€”â€”â€”â€” CHAPTER 5. â€”â€”
Section 5.3
2. Let be a solution of the initial value problem. First note that Differentiating twice, Given that
two equations give
3. Let and , the first equation gives
and
. and the last be a solution of the initial value problem. First write Differentiating twice, Given that
two equations give
4. Let and , the first equation gives
and
. and the last be a solution of the initial value problem. First note that Differentiating twice, Given that
two equations give and 5. Clearly,
and
converge everywhere. , the first equation gives
and
. and the last are analytic for all . Hence the series solutions 7. The zeroes of
are the three cube roots of
. They all lie on the
unit circle in the complex plane. So for
,
. For
, the nearest
________________________________________________________________________
page 194 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” CHAPTER 5. â€”â€”
root is , hence 8. The only root of is zero. Hence 9 .
. .
and
are analytic for all .
.
and
are analytic for all .
.
and
are analytic for all .
. The only root of
is . Hence
.
.
and
are analytic for all .
The zeroes of
are
. Hence
.
The zeroes of
are
. Hence
.
The zeroes of
are
. Hence
The only root of
is . Hence
.
.
and
are analytic for all .
.
and
are analytic for all . 12.
13 The Taylor series expansion of Let , about . , is . Substituting into the ODE, First note that
.
The coefficient of in the product of the two series is Expanding the individual series, it follows that Setting the coefficients equal to zero, we obtain the system
,
general solution is , ,
. Hence the , ________________________________________________________________________
page 195 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” CHAPTER 5. â€”â€” .
We find that two linearly independent solutions are Since and converge everywhere, 13. The Taylor series expansion of , about Let The coefficient of .
, is . Substituting into the ODE, in the product of the two series is
, in which . It follows that Expanding the product of the series, it follows that 1 Setting the coefficients equal to zero,
,
,
Hence the general solution is , , .
We find that two linearly independent solutions are ________________________________________________________________________
page 196 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” CHAPTER 5. â€”â€” The nearest zero of is at 14. The Taylor series expansion of Let Hence
, about , is . Substituting into the ODE, The first product is the series
.
The second product is the series Combining the series and equating the coefficients to zero, we obtain Hence the general solution is
.
We find that two linearly independent solutions are The coefficient
radius of convergence is analytic at , but its power series has a . ________________________________________________________________________
page 197 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”...
View
Full
Document
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

Click to edit the document details