,THEOREM 2.1 .
The solutions of the form
the
Dy=tg
DE are
to exponentials
't
The
✗e _
initial value problem Dy=7y with yeol yo
= has a
unique solution
y= yoett
Proof :
Dl ✗ ett ) =
✗ ☐ Ce
> ᵗ
= ✗ Jett
= I c- ✗ ett )
! .
✗ ett is a solution
let f- be solution to Dy=7y
arbitrary
:
an
ᵗf "
" "
Dce f) de ᵗDf
-
-
te
-
= -
" *
de f If
- -
= + e-
-
"ᵗ "t
Ife Ife
-
-
= -
= 0
i. All solutions are exponentials .
,THEOREM 2.2
→ THE DIFFERENTIAL EQUATION ( D-II ) " 0
y
: =
K LINEARLY INDEPENDENT SOLUTIONS { ett tett Hett -1k left }
-
HAS , , , . . . .
,
PROOF :
① Show that these functions are solutions
[ Mathematical Induction]
For base case : k= I [ Use Theorem 1.1 ]
For k=jtl ,
we have :
( D- II )Jᵗ
'
( tiedt ) = ( D-II ) ( D- II )J ( tie't )
CD II)J ( jti e' ᵗ + t.iett-7bje.tl
-
'
= -
+
( D- II )i( jtj
'
ett )
_
=
j CD II )iltJ- ett )
'
= -
= 0
ett II )J ]
'
ti [ CD
-
Ker
-
:
E -
i. tie't is a solution
② Show independent
that they are
linearly
" "" 't
i. tie t dzte tastzettt 1- 9kt
" '
0
-
=
. . . . e
> c-
i. e ( ✗ it 921-1-931-2 t .
. .
1- 9kt
" "
) = 0
.
_
.
ett =o or di 1- 921-1-931-2 t . . .to/4b-ktl-- O
Ma ↓
ett > 0 { I ,
t ,
-12 . .
.
tk -1 } is LI .
i. ai = 0
{ die 't ✗ zte 't A-stze.tt }
'
-
.
, , . .
.
is Lt .
, THEOREM 2.3
If
Yp and
Yg are non-trivial solutions to CD -
7pI)kPy = 0
and { D-
7g }kay=o { where 7q≠1p } then the set
{ yqiyp } is
linearly independent .
Proof :
We look for solutions for :
✗
ypt 1399--0
Apply CD 7pI ) :
-
i. CD -
7pF ) ✗
ypt Byq -0
CD 7pI ) Byq
:>
=0
-
BCD-7p-tlyq-oi.pc.IQ -
Ip )yq=0
Yp is non-trivial :
Ig =/ Ip
:>
13=0
Apply CD
Agt ) :
-
CD -19=1 ) typ tpyq =o
= .
✗ CD -79-+7 Cfp =o
✗
cap -2g )yp=0
= .
i. ✗ =o :
Xp -1-79
yp -40
The solutions of the form
the
Dy=tg
DE are
to exponentials
't
The
✗e _
initial value problem Dy=7y with yeol yo
= has a
unique solution
y= yoett
Proof :
Dl ✗ ett ) =
✗ ☐ Ce
> ᵗ
= ✗ Jett
= I c- ✗ ett )
! .
✗ ett is a solution
let f- be solution to Dy=7y
arbitrary
:
an
ᵗf "
" "
Dce f) de ᵗDf
-
-
te
-
= -
" *
de f If
- -
= + e-
-
"ᵗ "t
Ife Ife
-
-
= -
= 0
i. All solutions are exponentials .
,THEOREM 2.2
→ THE DIFFERENTIAL EQUATION ( D-II ) " 0
y
: =
K LINEARLY INDEPENDENT SOLUTIONS { ett tett Hett -1k left }
-
HAS , , , . . . .
,
PROOF :
① Show that these functions are solutions
[ Mathematical Induction]
For base case : k= I [ Use Theorem 1.1 ]
For k=jtl ,
we have :
( D- II )Jᵗ
'
( tiedt ) = ( D-II ) ( D- II )J ( tie't )
CD II)J ( jti e' ᵗ + t.iett-7bje.tl
-
'
= -
+
( D- II )i( jtj
'
ett )
_
=
j CD II )iltJ- ett )
'
= -
= 0
ett II )J ]
'
ti [ CD
-
Ker
-
:
E -
i. tie't is a solution
② Show independent
that they are
linearly
" "" 't
i. tie t dzte tastzettt 1- 9kt
" '
0
-
=
. . . . e
> c-
i. e ( ✗ it 921-1-931-2 t .
. .
1- 9kt
" "
) = 0
.
_
.
ett =o or di 1- 921-1-931-2 t . . .to/4b-ktl-- O
Ma ↓
ett > 0 { I ,
t ,
-12 . .
.
tk -1 } is LI .
i. ai = 0
{ die 't ✗ zte 't A-stze.tt }
'
-
.
, , . .
.
is Lt .
, THEOREM 2.3
If
Yp and
Yg are non-trivial solutions to CD -
7pI)kPy = 0
and { D-
7g }kay=o { where 7q≠1p } then the set
{ yqiyp } is
linearly independent .
Proof :
We look for solutions for :
✗
ypt 1399--0
Apply CD 7pI ) :
-
i. CD -
7pF ) ✗
ypt Byq -0
CD 7pI ) Byq
:>
=0
-
BCD-7p-tlyq-oi.pc.IQ -
Ip )yq=0
Yp is non-trivial :
Ig =/ Ip
:>
13=0
Apply CD
Agt ) :
-
CD -19=1 ) typ tpyq =o
= .
✗ CD -79-+7 Cfp =o
✗
cap -2g )yp=0
= .
i. ✗ =o :
Xp -1-79
yp -40
