Chapitre 2 : La résolution des équations différentielles

Transcription

Chapitre 2 : La résolution des équations différentielles
!
!
"
#
$
%
(
&'
)
*
*
!
"
"
%
(
#
+
+
+
&,
&.
-
.
.
+ *
.
,/01, 23
5 $/6 #
/-'7
0
4
8
9
:9
+&
/
+1α4&
&
$5
,
!
8
"!
#
/
"
%
;
(
#
# 5
<
& --
.
#$
** *
.
: *
.
.
* * =
# 5
%
.
.
$5*
>*
$
+
.
/
+
' () *+
?
*
!
#&
"
;
;
!
!
!
'
,
-
%.
#
-$
--
*
@
@
A
@
-
"
,
@
@
$
/
#
&
@ -
@
*
*
*
B
,
*
#
* #
C
** 5
*
;
--
#5
#
*
D
**
*
<
@
@ 4
--
#
-
<
-*
1$/ 4
**
@
*
--
** 5
*
E <
#
*
5
(
1
@
A
=
*
$- @
-
()
18 3
1
** B
*
-
--
*
=
)
#*
$/6 /
*
-$/6
*
<
$/6
#
$
"4@
$/
**
#
#
7
)
4
1
∀ ∈ ⊂
@
4
( )= (
( ))
*
#
7
1
4
1
4-
#
*
*
*
F
4
>
( )=
6
&
*
$
<
-
1$/64
<
-
$/
#
6
-
D
1
4
1#
*
&
-
*
-
-
-
#
-
-- &
*
<
7
<
*
*
#*
@
<
*
--
#
* -
#
*
*
**
<
*
5*
*@
B
*
*
*
*
5
7
< #
@
4
&
&
(
( ( ) ( )) =
-<
1
>
#
*
-
* 6 *
-
#
*
*
-5
5
*
*
=
@
*
#
*@
*
#
=
*
-
$/6 #
=
*
<
(
)
$/6 #
=
6
#
$/6
( )=
@
*
$/6
×
∀(
*
1
∈
# 5
1
4
1
4@
×
--
4
*
/
D
1
- G
=
( )=
&
4
7
) (
)∈
(
×
**
)− (
& *
1
H.%;I
4 >
> 5
−
1
4
A
&
<
J -
)≤
#
--
# *
# 5
&
*
/
$/6@
6
*
+
( )
** 5
−
=
*
*
*
#
#
( )
-5
( )
=
*
*
-5 @
# *
4
-
- G
6
<
*
#
&
( )
1
*
*
*
8
(
6
> #
6
( )
*
*
*
-
*
)
+
7
*
<
*
-
+
Φ
#
7
+ Φ(
=
)
+
-
1 !4
-
-
#
@
#
*
[
#
+
]
$
*
α
*
+α
+
α ≠
-
+α
+
7
(β
=
+ −
>@
+
+β
+
+ −
)
1 "4
< *
+
2 *
$
5
-
F
--
/
*
@
--
-
&
*
*
8
5*
* &
*
* *
-
#
*
#
*
**
@
#
#
--
5
*
5
#
*
@
*
* &
<
*
<
#
#-
*
*
@
#
#
7
#
5
-
@
F
*
% ! 0 %
* *
$
#5*
#
*
#
-
7
( )−
→
=
*
--
>
F
*
*
* *
5
@<
*
*
- G
7
<
!
** 5
*
*
: -
1 %4
**
#
)
<
<
*
*
#
@
F
<
* @
#
*
ε
*
7
+ Φ(
+ Φ(
=
+
=
+
6
)
1 ;4
)+ε
@
# 5
*
@
7
−
≤
−
*
*
7
ε
+
1
*
*
*
*
#
B
#
*
#
#
%
*
*
#
*
F
*
# *-
%
$- @
*
-
*
#
#
#
(4
--
*
7
<
*
#
**
#** 5
<
@
#
-
-
*
#
*G
*
<
* 7
(
=
+
6
( ))
1
4
1
4
7
=
→
+
Φ(
)− ( )−
@
A* @
D
-
-
* *
>-
*
D
*
#
7
- G
#
@
* *
*
-
7
1
# 5
4
A
#
#
6
7
≤
KE <
:
* *
#
@
*@
1
*
-
−
=
#
6
< #
*
#
4
&
#
&
"
@*
#
-
# *
** B
*
& * @
*
7
-
#
*
*
#
* *
H.%;I
@H
LM((I H
LM;!I*
@ #
&
*
"
* $
6
-
** 5
)≈
+
−
+
$
*
+
)≈
--
+
*
*
#
6
#
@
#
*
&
+
)
1
5*
4
@
N
#-
--
6
?
&
0
=−
*
*
&
*
5
#
*
*
4
>
*
*
6
+
5
# 5
=− *
*
*
#
#
5*
*
(
+
< *
*
#
=
5
@
*
#** 5
*
*
1
47
−
+
F
5-
5
)
*
*
1
)= (
(
+
*
** 5
-
=
+
< ** 5
+
&
7
( )= (
(
-
#
$
=
-5
--
*
7$
5*
O-
*
*
<
*P
#
*
&
F
8
@
&
#
*
*
=
*
=
#
$
** 5
%
#
#
#
0
−
=
5*
*
*
--
*
* 6
−
&
10
10
/
=
-
!%
"
@
*
*
*
*
7
*
*
*
#
A *
#
A
%
*
-
# *
<
−
*
+
@
=−
!%
0
6 *
*
=−
-
*
#
-
#-
/
-
*
#
*
#
<
< *&
-
&
*
@
7
*
-
#
-5
*
*
#
*
;
= @
@<*
*
*
*
*
*
> ** B
*
*
*
*
#
*
7
*
<
>
--
* @
# *
#
* *
*
#
--
*
*
*
F #
*
&
#
&
-
<
*
*
<
8
#-
**
*
*
>
--
<
*
#
F
*
#
# *
*
**
<
#
$/6
D
#
$
#
$/6 #
*
$/6 #
1
1
44
< #**
=
+
:
<
-
#
#
$
7
+
(
+
+
*
+
#*
"" @
=
+
*
-
*
#
(
<( @
**
*
1(
#
$
*
#
# *
@
4@
#
$
&
*
0 23
#
*
*
*
*
*
$
-- @
*
&
# *
#
0
<
*
-
-
0
A
*
F
#
:
* *
8
#
"4
@
D
#
$
/
*
#
F
* *
*
& F
$/6 #
*
*
1
F
)
#
#
$
#
+
* $
*5
*
!4
7
+
#
*
1
+
N
+
+
(
@
-
*
#
)
-5
*
&
5
*
*
10
/
1 %; 4
#
-
"
#
45 "
*
#** 5
*
60 % 7"
%
>
#**
#
C
-
*
7
+
% %
"
<
$/6@
10
!
*
0
#
!%
=
+
+
+
%
% %
!
0
89
%
+
<
*
1
*
#
$
+ -
@
#
**
7
%4
7%
=
=
+
+
=
+
1
+
#
* *
F
#
$- @ '
&'
*
#
*
*
*
*
1H
, ! I
47
+
*
-
**
-N
#
*
-
)
1 '$4 N
&
*
1 =4
*
5
*
5
'>
D
#
#
#
@
*
'$
C
'$ #
@
*
5*
Q
-- &
'$7
*
*5
-
-
*
*
-
7
-
1 '>
4
2 /
*
4@
#
#
C
)&
#
:
@
*
1
1 ':>
4 N>-
2 /
F
*
--
&
#
-
?
5 *
H.%;I $ - @
**
1 =4
:
--
** 5
#
#
F
*
**
#** 5
5*
J
=*
--
4
+
*
-
1
=
*
2
(4
∈
=
=
1
7
+
+
&
<
-
=
1 ;((4
*
-
-
L
#
1 ;( 4
#
;4
*
#
#
/ * @
$
@
@
&'
@
/3
−
−
+
$
@
@
L
+
RL
3
@
/3)
$
#
@,
--
•
#
≥ @ -
+
•
#
= @ -
+
H
, ! I
•
#
≥ @ -
+
H
, % I
* *
@
*
7
'$ #
#
*
#
*
' 1
*
**
*
"
#
+
4*
C
&
<@
#
<
* 5 N*
&
H
L "%I
7
#
7
< #
$ -- @ #
*
*
(
&'
S?
#
H
, ! I
?
*
5
-
*@
@
#
&
@
'>
#
1
*
<
#
?
**
*
#
0
#
*
#
*
*
=
<(
*
?
Q
@
#
#
λ
<
#
-
*
*
*
*
*
*
4 .
**
D
E
C
*
#5
*
1*
=
*
#
$
(
−
@
λ=
' *
*
$
*
−
*
(
*
'4 N
?
#
*5
@
5*
10
- $ --
#=
#
*
# 5%
*
&
*
7%
/3
:
/
@
#
*
*
<
#5
Q
- @
D
-
*
*
# *
#
*
< *
#
<*
$
**
#
-
#
*
&
*&
<
5
$
#
*
5( 6
5(
#
' #
5@
5T
*
*
*
*
5
*@ #
<7
=
J
#
*
+
<
1
−
1
4@
*
−
6 *
+
=
+
+
4
5
7
1
4
1
4
7
*
+
4
*
+
*
1
+
+
#
#
$- @ #
+
=
*
=
$
−
=
<
+
+
+
7
+
+
@
*
−
*
#
*
*
<
*
- G
#
4
1
!4
*
#
* 1
'
*
1
+
< #
−
−
+
#
**
+
+
** 5
=
*
−
+
−
=
#-
*
4 .
'
*
-
1 ; "4
*&
#
*
**
@
@
*
5
--
-
N
*
<
#
C
B
*
#
#
**
&
&
&
B
/
=
#**
--
*
N
* *
F
*
#
#
*
#
1 4
−
−
−
−
−
−
−
−
−
.
1 4
/
&
1 4
;
# *
#
#
*
1
*
5
4 #
-
$
7
-
*
-5
*
$
-- @
-- &
C @
> -=
$ <
*
/
7
-
@
*
*
*
-
@
7
1
#
&-
*
*F
*
*
# *
'@
*
E <
&
*
@
&
*
*
* *
&
*
<
*
*
'
*
*
*
5
(
&
*
*
*
E <
--
'*
# *
*
-- &
#
-
$
@
*
&
*
<
*
5*
**
*
#
5*
/ 3 @
-
7-
#
#
#
*
,
6 *
&
&
H
C ;%I
#5
-
#
* <
*
*
@
&
<*
*
$ # =
-
*
#**
*
*
*
-- &
*
* 7
*
2
*&
@
'H
C ;%I
D
#
*
-
*
β
α
--
1 "4 7
α
5&
+
+α
+
+α
*
*
!
%4
-
+
-
$
+
* *
*
+
≈
*
#
"4
* @
7
=
/
1
< @
*
*
−
=
=
+ −
=
(β
+
+
+β
+ −
)
@
=
/
--
α
1
*
4@
β
--
1
5
*
#
+
4 8
,/0
"
*
U +
<
<
-
#
#
+
#
*
)
-
*
*
@
** 5
5*
=
*
*
*
+
*
*
&
1
;4
+
#
6
#
+
-
5
=
*
#
< +
-
&
B
5
2(
*
)
*
#
*
−
! <
*
1
!4 >
*
@
)
#
−
+
10
7 ∇
=
)
$
γ ∇
γ = −
1
(4
- G
&
=
− +
*
)
7
−
!
−
+
− +
∇
*
* &
*
;4@
=
-
− +
10
/
+
--
*
6 *
)
** 5
*
*
*
-
7
+
$
0,
' )
#*
∇
<*
5
=∇
−
--
−∇
−
-
(
=
-
−
7
"
(
=
#
#
) (
−
1H
LM((I
4@
E <
#
*
#
*
−
#
+ γ
+
−
+
F
*
* *
F
>
*
#
#
8
-
C
*
--
& F
* *
*
?
#
=
+
=
+
=
+
=
+
=
+
−
−
+
=
+
=
+
−
−
+
5*&
* -5 N
*
-
+
−
*
<
=
%
*
#
$
&
*
* @
--
= @
*
-
'$
5*
*
**
*
*
5
,
7
8
**
N
#
<
#
+
4
# *
#
=
1
* *
#
#
#
/
*
-
*
−
"
*
−
−
(
−
−
9
2>
#
*
*
*
#
*
#
+
*
)
&
*
*
#
+
*
-
*
+
@
@
-
7
@
10
"4
#
<
*
,
#
*
%
*
γ =
+
-
* @
--
γ
--
7
γ + γ
*
)
− +
5
'>
7
!
− +
−
10
+
/
=
<*
γ ∇
+
=
--
γ
--
#
=
=
− +
#
+
,
+ γ
−
#
+
6
+
−
+
+
4
1
4
@
γ =
+
.
C
= @
*
= @
#
$
*
*
*A
=
+
=
+
=
+
=
+
=
+
=
+
+
+
−
=
+
=
+
+
+
−
"
*
>
#
+
+
+
%
−
+
−
−
9
@
#
+
*
-
1
1H
LM((I
47
γ + γ
*
>
γ = −
+
*
#
+
& )
F
5
+
−
− +
.
*
-5
S
>#
*
7
<
*
V
7
7
-
*
-
#
*
#
+
,
-
F
** 5
E <
** 5
*
=
+
+
+
+
&
7
+
7
V
*
#
+
+
*
=
+
+
+
+
;
6
$$
1
6 *
$4 N
+ #**
@
@ #
*
#
*-
&
&
+
*
-
*
<
*
$1
-5 1
$1 $484 *
W
--
4
&
#
>
(41<( %?@
*
-
#
*
*
E
*
%
=
--
#
+
5 ** 5
*
"′
0
1
*
*
"
)
4
=
+
6
− +
/ * @
>@ #
*
7
+
1
4
1
4
1
!4
"
- G
"
+
−
− +
+
− +
−
10
@
δ ∇
=
+
=
(
− +
δ = −
δ
7
*
*
7
=
,/0
#
+
-
+
δ =
#
"
.:
*
--
+
# *
≤
δ =
*
7
#
<
,/0
C
*
*
*
>
!
*
@
≥
#
*
*
!
*
,/0
*
&
=
+
=
−
=
+
−
+
+
=
+
−
+
=
+
−
+
=
+
−
+
=
+
−
+
'
=
−
+
−
−
−
=
−
−
−
+
−
−
−
−
−
−
−
+
−
−
=
−
−
+
+
−
−
=
+
(419
**
@
-
1
* #
#
W
@
4
8
# *
1
1
*
(
)
-
#**
"4
&
#
* &
#5*
8
*
1
#5*
*
&
*
*
*
*
7
44 > -@
*
# *
-
=
5*
+
+
45 "
$/6 #
*
=
−
%
>
#
+
#
*
#
'
*
*
:
*
:
/3
**
'1
=
+. A
4<1
4@
7
+
=
=
+
+
+
1
=
+
#
=
+
=
#**
1
%4@
%4
=
+
=
+
=
'<
$/6
7
$
&
=
+
+
+
+
=
+
=
+
+
$U 8
9
-
*
1
*
@
;4
-
(4
1
4
*
*
7
=
6
1
5
--
"4@
;4
X
*
1
1
=
*
*
+
% = $%
-#**
=
+
=
$= $
=
* #
@*
5
*
#
-
1
;4
=
+
+
+
=
+
8
=
+
+
9
*
$/6 #
1
W
>
4 +
H
LM((I 5
α
+ −
α
+ −
=
*
1
(4@
* , &
#
A
=
F
β
=
=
β
=
- G
'7
+ −
1
=
-
-
=
*
*
<
+
#
6 *
6 *
=
+
=
(
+ −
< 1
+ −
+ −
;4
)
=
5
@
*
-
*-
α
+ −
=
α =
=
#
7
β
=
αα−
4
=
(
+ −
+ −
)
1
β =
=
β β−
4
*
@
# *
$
*
* 1
:9
& @
−
+
)
<
--
=
#
+
−
+
−
−
−
+
−
C
Y
1
4
+
−
+
<
" *
−
=
−
-
:9
&Y
-
−
−
−
**
*
*
=
+
=
+
=
(
+
+
+
*7
)
+
(
+
+
+
1
4
--
&
)
-
8
-
#
O
P7
*
# *
/
*
**
6 *
*
*
#
*
@
*
7
?
#
/
*
<
*
#
*
#
@
+
H
L ;;I*
+
*
+
A
*
,
+
+
−
1H
L ;;I
4*
-
*
*
& >
-
4
** 5
:9
=
−
*
#
<
*
* :9
+
1
#
**
−
(
=
−
7
*
5*
+
**
$
*
+
1H
LM((I
4 $
(4 :9
+
>
*
**
*
−
-
5
N
*
*
@
-
*
*
-
5*
F -
* 1
<
*
*@
*5
*
#
*
*
&
*
4
*
* *
< (
&!
=
*
@ #
**
*
<(
@
<
-
*
*
@
*
5
@
# *
*
#
*
0
=−
=−
*
,/0
*
F
;
7
Z
*
#
1(
**
#
*
2[ > ** B
<*
*
5
*
$
*
5*
*
@
'@ $
5*
$
( "
( %*
&
'4
*
@
@
*
#
*
A *
*
/
@
5*
F
*
* *
* &
*
10
/3
B
4
0 %
(419 "
7
9
% 7
=−
-
*
#
>
N
*
-
# *
@
*
1H.%;I
4
<
@
1 5*
*
=−
!%
5*
*
*
4
% 7
*
-
*
5*
<
* <
&
7
-
@
6 *
*
&
# *
*
*
*
-
#
$/6-
*
1
<
*
--
4
*
@
*
X
*
&
*
*
-
7
+&
*2
+#
5
**
+&
&
*
*
*
@
-
U
:
#** 5
ϕ
**
-
** 5
--
∂
∂
<
@
= −ϕ @
1&
**
$ **
-
H
LM;!I7
( ϕ ( )) ( ) = & ( ) ( )
*
#** 5
4
=
-
( )=
1
#
#
$
1
*
U
4 / * @
U
@
7
=( + &)
1
*
+ λ
-
-
* *
<
*
!4
<
λ
"4
&@
*
#
&
7
( )=
=$
$∈
1
=$
*
A
%4
&
**
4
*
*
<
* @
*
--
&
-
&
<
+
**
/
='
1
;4
=$
*
'
5*
*
*
)
)
*
*
-
*
<
*
! (ξ ) = (α − β ) ξ +
*
*5 7
)
+ (α − β
)
1
*
*5
-
1*
4
* @
*
<
#
*
#
#
?
#
C 4
<
% % =9 <
0 =
0
'>
@
*
90 % 7/3
= *
0
;7*
"
<
7
$
5*
=− N
F
@"%
*
'$ #
*
**
J
<
#
* *
&
@
*
@
*
5
#
$- @
B
#
2
% 7
*
*
!
4
#
$ =− @
# *
<
'$@
1
6
#5
<
#
% 7
*
!ξ
-
'$ #
'>$
*
10
#
( *
*
*
<
)
(4
* 1H
LM;!I
4
0
*
@ #
1H
LM;!I
4
* @
#
<
'
*5
*
*
#
$
*
*
*
&
*5
D
* *
*
1
# *
%4
#
#
@
*
&*
6
+&:
'$
5
'>
+&
*
8
5 #
+
&,
0
-
5
*
=
*
#
+
&.
#
$
= @
N*
=
*
6
=
N*
*
*
#
*5 <
0
$
5*
@
*A
F
#* G
#
*
+&
#
+
&.
*
=
10
10
4
4
"
*
"
*
$
*
"
*
@
*
2(
2>
2(
?D 77
?D 77
E
!
?
,/0*
*5 1
@
0
4
*
=
@
*
+&
A
*
N*
#5
&
*<α
α=2
:
*
#
0
# * +&
@
$
*
,/0 #
*
@
*
#
"
10
4
*
(417?D 7
F7
'
+&
9
N*
@
-
7
(α
-
+1α4&
/
* 7
{
( −µ ) < α
(α = µ
α
0 %
*
1
+1;(\
4&
µ≠
-
}
<α <π
@
&
,/0
α
°
1
4N
+&
+1α4&
°
°
°
4
7
°
°
2
>*
* B
5
*
&*
]
*5 < *
5
'@
.
# *
:
5
*
J
-
1
7
'( )) = ∀)
6
-
→−∞
'(
)=
*
@
'(
→∞
)=
= ) )→∞
*
@
*
1
A1H
LM;!I
4
* *
%
;4 *
#5
'(
)
H
LM;!I
7
1
1
"4
O-
*
*
7
*
4
#5
P
4
(
=−
*
*
( )) ( ) =
−
*
<
0
#
$
*
*@
*
< #-
@
-
*
-
&
-
&
'(
=
*
=
< # =
#
→∞
$
*
*
-
@
F
@
10
+
4
=
=
*
*A
'
1
+&
7
)=
1
* -
*
4
*
&
% 0
+&
*
-
- G
6
1 **
=
#
$/6
*
N
A
(
/ * @
*
∀
<
*
+&
**
&
*
-
&
47
)− ( )
@
−
≤ν
−
#
$/6@
1
-
!4
7
;
−
ν≤
*
1H
LM;!I
4
ν≤
!4
@
* *
≥
*
,&
≤
@
-
−
1
*
# *
5
*
%4
*
* @
!
&
7
*
%4
"4
-
&'
*
1
5
#
*
∀ ≥
7
−
1
ν −
−
* *
#
$/6 6
1
≤
F
*
**
* *
-
*
D
<
-
,&
*
*
'6 *
H
LM;!I
* *
5
*@ $
-
*
#
*
,&
,&
+&
,/0 #
O<
*
P
(
> #
*
--
-
C
-
@
+1α4&
-
$ -- @
*
+&
*
* *
*
F
<
*
-
5*
&'
&,
-
+
2^
<
&
&
&
6
6
8
6
8
8
2_
#
*
-
+1α4&:
+&
2
$
*
C *A
+
&.
,/02[
,/02[
,/02[
/ * @
*
6
$
,&
&:
&
&
&
;(\
;(\
&
;(\
%!@( \
" @ \
,&
&
&
&
&
&
&
6
8
&
6
&
&
6
8
&
6
&
&
/%
C
*
(
>
%#
*
--
#
-*
*
5&
-
6
*
*
#
-
=
<
+ #**
#
$
@
#
*
*
#
<
5*
&
-
* *
!
"!
*
#
* *
# *
*
@
*
-
#
<
*
+
"!
"
4
9%
!
#
#
.
&
@
#5*
7
*-
J
*
#--
=
,
7$
*
<
*
10
*
%
<
#
$
-
'
-
!
&'
#
(( * 1
>
%
6
*
*
((( < ( (( K.C@
*
*
*
4
!
#
$
@$
--
"
*
0
=
#
=7/3
#
$
5*
H
,/ I
&
<
*
5*
=
#
#
* @
*
5*
K&
0 <
2
=
-
.
@
-
*
C
*
#
$
&
*
&
0
!
7
U* 4
-
< *
*
'
#
*
*
*
<
$
*
*
F
1< * &
?
1 '
4 6
*&
**
*
> 5
5N
1
5
*
*4
*
&'
*
&'
D
&
--
-
1H
L ;;I
47
+
10
'
G
7=
=
∀
=
1
. %
=
!% D G
7%= /3
!% D
%
G
;4
!% D
7 = /3
!% D
G
5
#5
*
-
'
*
<
S * *
-
5
#
*
*
* -&
@
. -
0
2
> 5
5
#
*@
@
#*
-*
F
7
*
**
$/6
@
#*
& --
*
*
&
*
@
<
*
"
*
1
4
-
-5
- G
−* − λ
=
λ
*
**&
6
-
&
" =
" − −λ
"
1 !(4
**
1 ;4@
$
<
*
6
7
*
*
*
@
#5*
@
#5
=
/
#5
@
+)=
N
+)−
)
*
* &
@
7
(
=
)
)− λ (
+) −
1 ! 4
7
=− λ
)= − − )λ
= +)−
-
#
$/6 `
*
*
1 -
D
1 ! 4
#
--
*
5
5
7
)=
#
5
*
--
7
- G
-
--
= @ a@
@
$/6
*
*
--
=ϕ
:
#*
-
=
λ
4
*
H
=
*
#5
* 1 ! 4@ #
-
$/6
=
6
#-
**
#
(
#
&
)=
*
#
"
#
< 47
(
*
1 ! 4
1
#
*
*
*
#
((I
*
#
*
>
>
.
%
&
J
L
-
N
*
#
=∂
@ **
* L
-
*
*
# *
@
+
# *
∂"
&
#
1 ;4 6
**
$
" =
@ ,
L
@
"−
""
+
@
-
#
&
1
(4
<7
=−
∂,
(
∂"
") " = −
∂,
(
∂
")
*
1 ! 4
*
O
PN
-
@
L
*
F
,=
*
*
#
-
#
1 ! 4
N
.
L
*
'= ′(
=
*
D
")
'&' = &
-
&=
,
-
*
1
#5
1 !!4
*
--
A
−
+
+
−
−
B
#
&'
*
@
C
;
−
7
*
− +
+
− −
+
+
.
&
#
*
−
7
>
$
;4
&
*
*
* *
--
−
**
"
*
7
<
#
&
* *
**
*
@ #
7
F
-
1H
LM((I
4
*
+*
*
7 #
*
=
%
H
IJCCK
--
#
-
6
0
#
1H
+
*
'$ #
* F
*
$ -
;"I
4@ # &<&
*
@
*
<
#
*
*
5
*
#
&
*
*
*
#
<
*
6@
*4
&
*
*
-
=
*
*
& F
*
*
-
-
--
&
*
-
#
1- &
6 *
F
J
*
* &
<
>
-
.
#
@
* *
O
#
P7
@
1
* − @
# *
*
#
*
-
**
*
4
-
*
*
&
*
*
@
--
↔−
#
1 !"4
#*
O
=
P
-
%
#
↔
5
< Φ
=
=
#-5
-
*@
&
=
#
#
$
5*&
@
*A
*
*
>
*
G%
*
* 7
&
D
&
*
#
(Φ )
*
Φ = Φ −−
Φ− =
Φ = Φ −− @
* *
%
&
F
*
* *
<
7
-
$
-
&
*
Φ
6
<
#
Φ !
D
&
*
<
-
#
*
*
<
*
#
*
=
D
<
>
*
*
-
=Φ
7
* =
1
*
4
*
--
N
&
-
*
* =
H A
; I
4
*
#
*
*
*
#*
-
1
*
H
L ;;I
@
#
#
$/6 >-
--
*
*
F
*
*
#5
5
5
*
*
H
L ;;I
5
@
*
* =
&
* =
5
@
*
*
-
@
#
7
7
5
*
*
7
*
<
@ K
* *
**
*
@
* =
*
&
*
5
>-
>-
E
*
* =
*
#
* =
*
>
*
* *
* -
&
*
/
%
"
#
5& @
-
5
& --
<
&
#
$/6
#
#
5 @*
1
#
/
5
5
* -
*
* * @
*
-
&
*
*
&
-
*
<
+-
*
-
&
4 6
7
"=#
" #= " # − - " λ
= "
"= "
#
5
" @ λ= λ
" @ #= "
:
-
@
λ
1 !%4
-=∂
*
∂"
#
5
7
"
-"
/
- G @
*
#
$/6
#
<@ #
#
#
F #
#
λ
=
λ
5*
"#
− "" # #
"
#
@
!
#
-"
5
1 !;4
# >
*
*
*
*
7
#
1 !;4
-
*
#
<
@
+-
<
<
@
*
.
#
< #
5
1 !%4@
@
*
-
-
*
*
*
@
#**
&
*
*
:
*
*
*
**
6 *
*
@
F
*
5*
5*
8
G
*
*
#
*
@
>
*
*
X
*
*
#
*
@
#
$
1H ; I
4@ #
$
1H
/K;!I
@H
, ((I
@ H /+((I
4@
<
$
%
?
*
&
5*
#
-
* &
%
/
5*
@
-
* =
#$
Y
1 !;4
@
5
** B
5
*
@
*
* =
-
-
7
-
*
5
<
@
H
+(; I
@
5
&
*
*
--
*
* =
-
*
/#
-
*
-
'$1 ' *
H
Y .; I
4 /
#
< #
1HU
0; I
4@
:9
&
H
, ; I
@H
U ; I
@*
#=
-
@
*
* @
--
5
D
<
*
*
#
#*
=
-
*
*
*
*
-
#
#
- G @
@
$
@
<
* *
@
*
> #
#-# *
+
*
7
-
=
1H ; I
4
S
#
=
*
* b
@
&
*
-
*
#
D
=
-
-
< #
*
#
N
5
*
F
&
-
#
/
#
#
*
*
#
&
*
#
&
"
*
#
#
*
*
#
@
*
@
$
#
=
-
--
*
* &
**
*
#
# *
--
*
#
*
#
* + U 2
#
@
*
* *
-
$
*
@
*
<
*
2I
#
*
1
4
-
* <
*
*
*
-
&
4 /
F @
--
*
*
#
H
/K;!I
@
-
*
*F
#
*
5
*
#
1
#-
/#
*
<
H
L
*
*
D
*
=
*
&
-
#*
#
@
#
*
#
N
E <
*
* &
*
>
*
#
&
H
U ; I* *
&
7
*
<
* *
@
#
<
&
* <
@
<
*
1H
C ,0%"I
4
&
-
#
E <
5
*
*
--
$/6
*1
*
-
*
*
$ -- @ #
-
**
*
P
J
*
O
-
*
&
#
*
&
< #
-P
-
&
*
-
<-
-
$
%
5*
*
#
$/6
#*
4 >
*
O 5*
*
7
+∆
=
(∆
)
1 "(4
J
*
#
*
4 $
-
<
1
4
< #
**
--
<
*
*
-
+∆
< #
J
F
*
5*
-
*
*
#
>
*
--
< -
*
*
--
#
$
#
* *
*
=
< --
*
--
-&
-
?F
1 "(4@
#
$
#
*
$
@
?F
*
-
**
@
*
#
F
5*
#
#
*
&
*
*
< #
*
#
6 *
< #*
*
*
5*
*
** 5
-- @
+ -
+∆
* < #
*
*
<
*
$
-
#
< #5*
*
#
*
#
$/6
#5*
$/6
;
-
*
>
*
*
#-
*
--
1
*
>
)
*
4
%
*
5*
-
*
1W
1
O
4@
-
P 8
*
*
8
#
*
*
< #
@
#
G
*
#
$
*
#
#
--
M 2
*
H
,M;%I
#
< #
Z
# *
@*
-
, &
&
7
=
;
7 #
−
−
∂
+
∂
@
5
5*
⇔
=
1 " 4
*
51
* &
H
LM;!I
4
;
J$ *
#*
#
-
0
#5*
*
$+
$ *
<
$/6
&
5*
* *
#
$
-
5*
*
@
#
*
*
#
>
#
*
<
<
*
=
+
+
**
#
7
+
@
1
&
+
1 " 4
+
--
+
** 5
C
-
4 /
<
- G @
7
−
−
∆
*
∂
−
∂
∂
∂
−
-# *
-
<
-
#
2
#
*
7
5
7
#-
# *
2
@ D@ a4
*
@
&
>
<
7
#
@
>
#
--
5
&
*
*
*
1K
@
#
- G
*
2
&
&
*
-
*-
<
-
*
7
>
+
*
=
*
&
*
8 3
4
6
-
+
=
*
1 " 4
-
5*
<
*
∂
∂
+
*
-
1
−
∆ =
1
*
**
-
--
4
<
. &
*
&
5
5*
*
#5*
<
5
*
-
*
!(
*
7
K
=
@*
&
>
-
1 " 4
*
*
7
**
#
*
@ #
*-
--
*
/ * @
*
# #
*&
=&
-
*
<
** 5
*
@
--
F
*
--
* *
*
=
-
*
?
*
*
*
#
*
*
5&
&
@
F
*
#
*
*
#
*
*
*
< #
•
#
*
7
*
•
-
•
#
-
-
7
*
=
@
*
4 >
#
1
#
*
*
H
,M;%I : 9
*
&
*
5
>
-
<
*
*
**
-
H
,M;%I
*
*
*
?7
<
*
*
&
*&
5*
*
#
&
<
@
*
*
*
P
* < * *
*
&
* -
7
O
*
#
&
5
#
*
* @
@
*
**
N
#**
# *
*
&
*
**
*
? =
H
:/,;;I
S
*
>@
&
5
D
$
-
5
<
?
N
-
5
&
>
*
?
*
5
#
*
#
>
#
*5
#
$
&
<
5
#
--
7
&
*
!
4
%L
% %
H
$$L((I
*
* *
5*
7
*
O
*
@
*
P
E <
1
* *
O
4
#
#
*
*
P $
-- @
*
+
5*
&
1H
+ M; I
4@
&
*
-
*
@
S
*
<
*
*
?
*
&
@ #
>
*
F
*
*
&
>
?
F
@
*
# * &F
<
*
&
&
*
H
,M;%I >-
S
-
*
?
@ <
H
,M;%I
<
*
/*
*
*
*
>
-
*
# *
*
*
&
,/0
5*
*
%
8
*
8
*
*
*
#
$
*
*
:
&
*5
*
@*
*
5
* *
*
F
-
*@
*
*
#
$
*
*
(
@
@
&
#
+ &
<
O
P@
-
*
*
=
#
#
*
H
:/,;;I*
-
$ -- @
*
*
#
**
*
H
Y.((I* *
&
#
@
=
F
*
+&:
,
* H
Y.((I
-
*
-
@
#
N
*
6 *
@
*
*
*
@
' () *+
8
*
* -
-<
*
-
5
-5
7
*
@
8
!
5
&
* *
@
*
5
&
*
*
# -
@
5
&
'
* *
*
*
<
>*
*
#5*
<
O
*
1$ &
*
-
#
c$5
<
P
* *
*
$ -- @
>-
# *
&&
1$ &
*
*
*
=
c
<
*
--
* *
*
5
&&
c
4
#c$ &
* &
*
*
c
4
# *
5*
<
8
*
X
*
*
*
-
* #
#5
57 #
*
*
/ * @
#
-
5*
<
5*
*
*
*
*
<
*
X
&
:6 $
!
!

Documents pareils