ED linéaires d`ordre 2 à coefficients constants

R C
L q i =dq
dt
q
C+Rdq dt+Ld2q
dt2=U0sin(ωt)
d2q
dt2+ 2λdq dt+ω2
0q=U0
Lsin(ωt) (ϕ)
λ=R
2Lω0= (√LC)−1λ
ω0
λ= 0 U0= 0 q ω0(ϕ)
y00 +ay0+by =f(x) (E),
(a, b)∈K2f:R→K

(E)R K y:R→K
∀x∈Ry00(x) + ay0(x) + by(x) = f(x)
(E)
y00 +ay0+by = 0 (H).
z1z2(E)R(z1−z2)
(H)R
SESH(E)
(H)z∈ SEz
SE={z+y, y ∈ SH}.
SESH
z1z2(E)R
z1−z2x∈R
(z1−z2)00(x) + a(z1−z2)0(x) + b(z1−z2)(x)
=z00
1(x) + az0 1(x) + bz1(x)
| {z }
=f(x)
−z00
2(x) + az0 2(x) + bz2(x)
| {z }
=f(x)
=f(x)−f(x)=0.
z1z2
(E)z1−z2(H)
z(E)SE={z+y, y ∈ SH}
Γ
•g∈Γy(H)g=z+y g
Rx∈R
g00(x) + ag0(x) + bg(x)=(z+y)00(x) + a(z+y)0(x) + b(z+y)(x)
=z00(x) + az0(x) + bz(x)
| {z }
=f(x)z∈SE
+y00(x) + ay0(x) + by(x)
| {z }
=0 y∈SH
=f(x) + 0 = f(x),
g∈ SE
•˜z∈ SE˜z−z(E)
(H)
˜z=z+ (˜z−z)
| {z }
∈SH
,
˜z∈Γ

(E)
•(H)−→
•(E)−→
SH
y1y2(H)Rλ µ
K(λy1+µy2) (H)R
SH
y1y2(H)λ µ Ky1y2
λy1+µy2
(λy1+µy2)00 +a(λy1+µy2)0+b(λy1+µy2)
= (λy00
1+µy00 2) + a(λy0 1+µy0 2) + b(λy1+µy2)
=λ(y00
1+ay0 1+by1
| {z }
=0 y1∈SH
) + µ(y00
2+ay0 2+by2
| {z }
=0 y2∈SH
)
=λ·0 + µ·0=0.
SH
2
2
a b
y00 +ay0+by = 0 (H)
r∈Cu:R→C
x7→ erx
uRu0=ru u00 =r2u
∀x∈Ru00 +au0+bu = (r2+ar +b)u
u:x7→ erx (H)r2+ar +b= 0

(H)
x2+ax +b= 0,
a b (H)
r1r2r1+r2=−a r1=r2
−a/2
x2+ax +b= 0 (H)
•r1r2Cy1:x7→ er1xy2:x7→ er2x
(H)
•r∈Cy1:x7→ erx y2:x7→ xerx (H)
(a, b)∈C2(H)y00 +ay0+by = 0
∆x2+ax +b= 0
SC
H(H)
•∆6= 0 Cr1r2
SC
H=x7→ λer1x+µer2x,(λ, µ)∈C2.
•∆=0 Cr
SC
H=x7→ (λx +µ)erx,(λ, µ)∈C2.
Γ = x7→ λer1x+µer2x,(λ, µ)∈C2et e Γ = x7→ (λx +µ)erx,(λ, µ)∈C2.
⊃
•∆6= 0 r1r2
y1:x7→ er1xy2:x7→ er2x(H)
λy1+µy2(H)SC
H⊃Γ
•∆=0 r
y1:x7→ xerx y2:x7→ erx (H)
(λy1+µy2) : x7→ λxerx +µerx (H)SC
H⊃e
Γ
⊂
y∈ SC
H(H)r

z:x7→ e−rxy(x)z
u:x7→ erx
y=uz,
y0=u0z+uz0,
y00 =u00z+u0z0+u0z0+uz00.
y(H)⇐⇒ y00 +ay0+by = 0
⇐⇒ (u00z+u0z0+u0z0+uz00) + a(u0z+uz0) + buz = 0
⇐⇒ uz00 + (2 u0
=ru +au)z0+ (u00 +au0+bu
|{z }
=0 u∈SC
H
)z= 0
⇐⇒ uz00 +u(2r+a)z0= 0
x u(x) = erx 6= 0
u
y(H)⇐⇒ z0Y0+ (2r+a)Y(∗) = 0.
y(H) (∗)λ∈C
∀x∈Rz0(x) = λe−(2r+a)x.
•∆6= 0 r1r2r=r2
r1=−r−a r16=r22r+a6= 0 z0z µ
∀x∈Rz(x) = −λ
2r+ae−(2r+a)x+µ,
∀x∈Ry(x) = erxz(x) = −λ
2r+a
| {z }
:=˜ λ
e(−
2r−a+ r)x+µerx =˜
λer1x+µer2x.
y∈Γ
•∆=0 r2r+a= 0
z00 = 0.
Rz0z λ µ
∀x∈Rz(x) = λx +µ,
∀x∈Ry(x) = erxz(x) = (λx +µ)erx.
y∈e Γ