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OMPI 3 - Équations différentielles
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Condensé du cours de OMPI abordant les principaux thèmes de la matière comme : Equations différentielles, calcul différentiel, dérivation...
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^OMPI 3 - ÉQUATIONS DIFFÉRENTIELLES
I. Équations différentielles linéaires à
coefficients constants
e , , , , .gg, ça , .am ,
,
nappe, ,
.my, gym, ,, ,
.
, ,, ,
, , .
avec xlo) =
% avec Ca , %) C- À forcés est :
x ( t) = xoeat .
•
Rappel : Considérer l' équation différentielle linéaire d' ordre 2
{&¥H+w2x(Ê( 4=0 Posons
y
/
1-1=1%7 ¥11 Ayltl -0
-
, +
xlo ) =
Roi c) = va avec d- =
¥; ! E IR
"'
et
y /c) =
( Y ) C- R2
( )
ûltl âltl
Donc
0¥ (a) =
ült )
=
www
=
Aylt )
"
•
Théorème : Soit A E Rd et y C-
Ri . L' unique solution
C- l' Ko + •
[ Rd ) de l' équation différentielle telle
que
-
y ; ,
.
9¥14 AHN et
"
donnée
ylt )
=
est
par
=
Yo
ylo ) =
Yo
où on a EB =
ËÎ B
"
€ Rᵈ×ᵈ si BERN .
"'
•
Remarque : Sert BE Rd une matrice carré . Vu
que
F- =
Rᵈ×ᵈ et de clim
finie d
'
et est donc un
espace
de Banach par toutes les normes .
On définit sur E
la norme :
111311 IB / "
où euclidienne
I. l aile
,
=
sup naine
✗ ERIK} IN sur 112¢
Cette d'
Algèbre B Bz sert
-
norme en une norme : si ,
et
dans F- .
IIB Ballz,
≤ 1113,11211132112
En particulier, tn C- IN ,
ABEE ; 1113^112 ≤ IIBHÎ
Donc la série Ë 1- Bn et normalement
na n!
convergente car :
c- D + A
"☐"
¥ KI:B
"
", ≤
n
? ¥111311 ? = e
2
< es
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