*This preview shows
pages
1–3. Sign up
to
view the full content.*

——————————————————————————
——
CHAPTER 4.
________________________________________________________________________
page 146
Chapter Four
Section 4.1
1.
The differential equation is in standard form.
Its coefficients, as well as the function
1> œ>
ab
, are continuous
everywhere
.
Hence solutions are valid on the entire real line.
3.
Writing the equation in standard form, the coefficients are
functions with
rational
singularities at
and
.
Hence the solutions are valid on the intervals
, ,
>œ!
>œ"
_!
a
b
!"
"_
,
, and
,
.
4.
The coefficients are continuous everywhere, but the function
is defined
1> œ68>
and
continuous only on the interval
,
.
Hence solutions are defined for positive reals.
!_
5.
Writing the equation in standard form, the coefficients are
functions with a
rational
singularity at
.
Furthermore,
is
, and hence not
Bœ"
:Bœ>
+
8B
ÎB"
!%
a
b
undefined
continuous, at
,
Hence solutions are defined on any
B œ„ #5" Î# 5 œ!ß"ß#ßâÞ
5
1
interval
does not
that
contain
or
.
BB
!5
6.
Writing the equation in standard form, the coefficients are
functions with
rational
singularities at
.
Hence the solutions are valid on the intervals
,
,
Bœ„#
_ #
a
b
# #
# _
,
, and
,
.
7.
Evaluating the Wronskian of the three functions,
.
Hence the
[ 0 ß0 ß0 œ "%
"#$
functions are linearly
.
independent
9.
Evaluating the Wronskian of the four functions,
.
Hence the
[ 0 ß0 ß0 ß0 œ!
"#$%
functions are linearly
.
To find a linear relation among the functions, we need
dependent
to find constants
, not all zero, such that
-ß-ß-ß-
-0 > -0 > -0 > -0 > œ!
""
##
$$
%%
.
Collecting the common terms, we obtain
a
b
- #- - > #- - - > $- - - œ!
#
$%
"$%
"#%
#
,
which results in
equations in
unknowns.
Arbitrarily setting
, we can
three
four
-œ"
%
solve the equations
,
,
, to find that
,
- #- œ " #- - œ " $- - œ "
- œ #Î(
#
$
"$
"#
"
- œ "$Î( - œ $Î(
#$
,
.
Hence
#0 > "$0 > $0 > (0 > œ !
"
#$%
.
10.
Evaluating the Wronskian of the three functions,
.
Hence the
[ 0 ß0 ß0 œ"&'
functions are linearly
.
independent

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*——————————————————————————
——
CHAPTER 4.
________________________________________________________________________
page 147
11.
Substitution verifies that the functions are solutions of the ODE.
Furthermore, we
have
[ "ß -9= >ß =38 > œ "
ab
.
12.
Substitution verifies that the functions are solutions of the ODE.
Furthermore, we
have
.
[ "ß>ß-9=>ß=38> œ"
14.
Substitution verifies that the functions are solutions of the ODE.
Furthermore, we
have
.
[ "ß>ß/ ß>/
œ/
>
>
#>
15.
Substitution verifies that the functions are solutions of the ODE.
Furthermore, we
have
.
[ "ßBßB
œ'B
$
16.
Substitution verifies that the functions are solutions of the ODE.
Furthermore, we
have
.
[ Bß B ß "ÎB œ 'ÎB
#
18.
The operation of taking a derivative is linear, and hence
-C -C
œ-C -C
""
##
"
#
"#
5
55
.
It follows that
P-C -C œ-C -C : -C
-C
â: -C -C Þ
cd
‘
"
#
" "
#
8 ""
"
#
88
8
"8
"
a
b
a
b
Rearranging the terms, we obtain
Since
and
P-C -C œ-PC -PC Þ
C
C
c
d
"
"
#
#
"
#
are solutions,
.
The rest follows by induction.

This is the end of the preview. Sign up
to
access the rest of the document.