Équations di érentielles.

•
•
•
R
0
0( ) = ( ,( )) ∈,
0= ( ,).

1) 0= + ( ,) = +
2) =]0,+∞[20= + 0=+
2
2(,) = +
2
∈0( ) = ( ,( ))
R( ) = 2+R
∈R0( ) = ( ) + 2 −2=R R
0= + 2 −2
(,) = = R
0=R.
( ) = α
α∈Rα
( ) = α α ∈R
0= 0 2
( ) = α(0) = 2 (0) = 2 α= 2
( ) = 2 ∈R.
UFR S T B
ST
UFR

(0,)
0∈ ∈ R
(0,) ( 0) =
(0) =
(0,2) (0) = 2 (0) = 2
0=R.
(0) = 2
0= ( ,)
(0) = ( 0) =
0= ( ,)
(0) =
,
( ) 0+ ( ) = ( ) ,
( ) 6= 0 ∈
0=−( ) + ( )
( )
(,) = −( ) + ( )
( ) (3.1)
UFR S T B ST
UFR

20−=√=]0,+∞[
( ) 0+ ( ) = ( ) ( ) = 2 ( ) = −1
( ) = √
( ) 0+ ( ) = 0 ,
( ) 0 ( ) = 0 ∈
( )
0
=−( )
( ) | |0
=−0( )
| | =−( ) + | | =−( )
λ=λ=−=λ−( ) λ∈R∗
( ) ( ) ( ) = λ−( ) λ∈Rλ= 0
( )
•
7→ ( ) ( )
0( ) = ( )
( )
∀ ∈
=λ−( ) λ∈R
(0) =
UFR S T B
ST
UFR

1) ( 0)=0
(0)=0
2)
λ λ ∈R
3)
3)
(cos ) 0+ (sin ) = 0 =] −π
2,π 2[
( ) 0
( )
0=−sin
cos ln | | = ln |cos |+
| | =|cos |=λcos λ∈R∗λ=±=λcos
λ∈R
( ) 0+ ( ) = ( ) 0
+0
−0
( ) 0+ ( ) = ( ) ,
( ) 0+ ( ) = 0 ,
UFR S T B ST
UFR