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TSI2
a(t)y00(t) + b(t)y0(t) + c(t)y(t) = h(t) (E)
a, b, c h Ra
y(E)t
Y(t) = y(t)
y0(t)
Y(S)
Y0(t) = A(t)Y(t) + H(t)
A(t) = 0 1
−c(t)
a(t)−b(t) a(t)!H(t) = 0
h(t)
a(t)!
(H) (E)
(u, v) (H)t
U(t) = u(t)
u0(t), V (t) = v(t) v0(t)
(S)
W(t) = λ(t)U(t) + µ(t)V(t)
λ µ C1R
W(S)tR
λ0(t)U(t) + µ0(t)V(t) = H(t)
t
(λ0(t)u(t) + µ0(t)v(t) = 0
λ0(t)u0(t) + µ0(t)v0(t) = h(t)
a(t)
y:t7→ λ(t)u(t) + µ(t)v(t)
(E)
X
p>0
(−1)pt2p

X
p>0
(−1)pt2p+1
(Ec) : y00 = 0
(H) : (1 + t2)y00(t)+4ty0(t)+2y(t)=0
f(H)R
t7→ (1 + t2)f(t)t
t7→ 1
1+t2, t 7→ t 1+t2(H)
(H)
(H) 0
y(t) =
+∞
X
n=0
antn
an, n ∈N
n∈Nan+2 an
p a2pa2p+1 p, a0a1
+∞
X
n=0
antn0
(E)
(1 + t2)y00(t)+4ty0(t)+2y(t) = 1
1 + t2
(E)
t7→ y(t) = λ(t)
1 + t2+tµ(t) 1 + t2
λ µ
t λ0(t)µ0(t)
tλ0(t) = −t
1+t2
µ0(t) = 1
1+t2
t λ(t)µ(t)
(E)
H
n∈N∗In0nMn(R)
A∈ Mn(R) ∆, N ∈ Mn(R)
A= ∆ + N
∆
N p ∈NNp= 0n
∆N=N∆ ∆ N

TSI2
(∆, N)A
A1=3 2
0 3 A2=3 1 0 2 
∆ = 3 0
0 3 N=0 2 0 0 (∆, N)
A1
∆ = 3 0
0 2 N=0 1 0 0 (∆, N)
A2
A3Mn(R)A3
A4Mn(R)A4
(∆, N)A∆N A
M2(R)
n= 2 AM2(R)
χA(x) = x2−(tr A)x+ det A
(tr A)2−4 det A > 0A
(tr A)2−4 det A= 0 A
A, I tr(A)
(tr A)2−4 det A < 0A
A= ∆ + N
∆N∆N=N∆p∈N∗Np= 0
λ N X ∀k∈N∗, NkX=
λkX
0N
P∈GL2(C)α∈C
P−1NP =0α
0 0 
N2= 0
λ∈R∆X6= 0
A(NX) = λ(NX)
λ A
M3(R)
A=
3−1 1
0 2 2
−113
∆ = 
2 0 0
−131
−113
N=
1−1 1
1−1 1
000
∆
∆ ∆
∆P−1∆P=D P ∈ M3(R)
D=
200
020
004
.

∆−1∆−1P
P−1D1
A
A
N
α∈RN∆=∆N=αN
(∆, N)A
A−1
N1= ∆−1N
∆−1N=N∆−1
N1
(I3+N1)(I3−N1)I3+N1
A−1
A−1= (I3+N1)−1∆−1A−1
A
p∈N
(∆, N)A
Ap= ∆p+p2p−1N.
(∆p, p2p−1N)Ap
∆p
Ap= 2p−1
2 + p−p p
1 + p−2p1−p+ 2pp−1+2p
1−2p−1+2p1+2p
R R2=A R A
U∈ M3(R)U2=D
P P −1S
S2= ∆
a b S =a∆ + bI3
S N1
M=I3+1
2N1M2=I3+N1
A= ∆I3+ ∆−1NR R2=A
R
H

TSI2
N
y(E)a
∀t∈R, y00(t) = h(t)−b(t)y0(t)−c(t)y(t)
a(t)
Y0(t) = y0(t)
y00(t)= y0(t)
h(t)
a(t)−b(t) a(t)y0(t)−c(t) a(t)y(t)!= y0(t)
−c(t)
a(t)y(t)−b(t) a(t)y0(t)!+ 0
h(t)
a(t)!
= 0 1
−c(t)
a(t)−b(t) a(t)!y(t)
y0(t)+ 0
h(t)
a(t)!
Y(S)Y0(t) = A(t)Y(t) + H(t)
A(t) = 0 1
−c(t)
a(t)−b(t) a(t)!H(t) = 0
h(t)
a(t)!
(H) : a(t)y00(t) + b(t)y0(t) + c(t)y(t) = 0 (E)
a
(H) : y00(t) + b(t)
a(t)y0(t) + c(t) a(t)y(t)=0
(H)R2
(u, v) (H)t
U(t) = u(t)
u0(t), V (t) = v(t) v0(t)
U V (S)
U0(t) = A(t)U(t)V0(t) = A(t)V(t)
W(t) = λ(t)U(t) + µ(t)V(t)
W0(t) = λ0(t)U(t) + λ(t)U0(t) + µ0(t)V(t) + µ(t)V0(t)
=λ(t)U0(t) + µ(t)V0(t)+λ0(t)U(t) + µ0(t)V(t)
W(S)⇐⇒ W0(t)−A(t)W(t) = H(t)
⇐⇒ λ(t)U0(t) + µ(t)V0(t)+λ0(t)U(t) + µ0(t)V(t)
−λ(t)A(t)U(t)−µ(t)A(t)V(t) = H(t)
W(S)⇐⇒ λ(t)U0(t)−A(t)U(t)
| {z }
=0
+µ(t)V0(t)−A(t)V(t)
| {z }
=0
+λ0(t)U(t) + µ0(t)V(t)=H(t)