Équations Différentielles Linéaires et Suites Récurrentes

RC R
C
Ecos(ωt)t= 0
u(t)
KR C I
R
u(t) + Ri(t) = Ecos(ωt).
i(t) = Cdu
dt =Cu′(t).
u
RCu′(t) + u(t) = Ecos(ωt).
u

a b I K
y′(t) + a(t)y(t) = b(t),
y:I→K
∀t∈I, y′(t) + a(t)y(t) = b(t).
y′(t)y(t)
y′(t)2+ey(t)t2= ln(t)
y
y′
y′(t)+a(t)y(t) = b(t)
C1I y′(t) = b(t)−a(t)y(t)
y′(t)−2ty(t) = t3
y′−2ty =t3
y t
y′−2xy =x3
c(t)y′(t) + d(t)y(t) = e(t)
c(t)
c
y′(t) + d(t)
c(t)y(t) = e(t) c(t)
ty′(t)−y(t) = 1
tR∗
+R∗ −
t
y′(t)−1
ty(t) = 1 t2.

b
y′(t) + a(t)y(t)=0
(E) : y′(t) + a(t)y(t) = b(t)
y′(t) + a(t)y(t)=0
(E) (E0)
(E) : y′(t) + a(t)y(t) = b(t)
(E0)
S={y∈ D(I, K)| ∀t∈I, y′(t) + a(t)y(t) = b(t)}
(E)SH
(E0)
SH
(u, v)∈ S2
H
(λ, µ)∈K2
λu +µv ∈ SH.
D(I, K)
I K
f
t∈I f′(t) + a(t)f(t) = 0 + 0 = 0 f∈SH
u, v ∈SHλ, µ ∈K λu +µv I
t∈I
(λu +µv)′(t) + a(t)(λu(t) + µv(t)) = λu′(t) + µv′(t) + λa(t)u(t) + µa(t)v(t)
=λu′(t) + a(t)u(t)+µv′(t) + a(t)v(t)= 0.
λu +µv ∈SH
yp∈Sp
y∈ D(I, K)y−yp∈SH
0
Sp={yp+yH|yH∈SH}

y∈ D(I, R)t∈I
(y−yp)′(t) + a(t)(y−yp)(t) = y′(t)−y′
p(t) + a(t)y(t)−a(t)yp(t)
=y′(t) + a(t)y(t)−y′
p(t) + a(t)yp(t)=y′(t) + a(t)y(t)−b(t).
y′(t) + a(t)y(t) = b(t)⇐⇒ (y−yp)′(t) + a(t)(y−yp)(t)=0.
y∈S′⇐⇒ y−yp∈SH.
∃yH∈SHy−yp=yH⇐⇒ y=yp+yH.
S′={yp+yH|yH∈SH}.
(E) (E0)
yp
b1b2
I y1(E1) : y′(t) + a(t)y(t) = b1(t)y2
(E2) : y′(t) + a(t)y(t) = b2(t) (λ, µ)∈K2λy1+µy2
y′(t) + a(t)y(t) = λb1(t) + µb2(t).
t∈I
(λy1+µy2)′(t)+a(t)(λy1(t)+µy2(t)) = λ(y′
1(t) + a(t)y1(t))+µ(y′ 2(t) + a(t)y2(t)) = λb1(t)+µb2(t).