Notes de cours: Les équations di érentielles. I) Equations di

y0=f(t, y) = a(t)y+b(t) (E)
a b I R
a(t)b(t) (E)
y0=a(t)y+b(t)
b(t)=0 ∀t∈I.
(E)y0=a(t)y+b(t)
y0=a(t)y(Eh)
y1y2(E)y0=y1−y2(Eh)
(E) (Eh)
(E)
(Eh)I
y(t) = keA(t)
k A a I
(E)y0=a(t)y+b(t)
I
y(t) = keA(t)+l(t)eA(t)
k l b(t)e−A(t)

•b(t) = est
y(t) = λest s6=a
y(t) = λxest s=a
•b(t)n b(t) = P(t)
y(t) = Q(t)Q(t)
n a 6= 0
n+ 1 a= 0
•b(t) = estP(t)P(t)n
y(t) = estQ(t)Q(t)
n s 6=a
n+ 1 s=a
•b(x) = b1(x) + b2(x)
y1(E1) : y0=a(x)y+b1(x)y2
(E2) : y0=a(x)y+b2(x)y1+y2
(E) : y0=a(x)y+b(x)
y00 +a(x)y0+b(x)y=c(x) (E)
a b c I Ry
x I
x I y y(x0) = y0y0(x0) = y0
0
y0y0
0
y00 +a(x)y0+b(x)y=c(x) (E)
c(x)=0 x I
(E)
y00 +a(x)y0+b(x)y= 0 (Eh)

y1y2(E)y1−y2(Eh)
(E) (Eh)
(E)
y00 +ay0+by =c(x)
a, b c I
Ry x I
(E)
r2+ar +b= 0
∆ (E)
•∆>0r1r2
(Eh)
y(x) = α1er1x+α2er2x
α1α2
•∆ = 0 r0
(Eh)
y(x) = α1er0x+α2xer0x
α1α2
•∆<0r1=u+iv r2=u−iv v 6= 0
(Eh)
y(x) = eux (α1cos vx +α2sin vx)
α1α2
•c(x) = esx
y(x) = λesx s
y(x) = λxesx s
y(x) = λx2esx s

•c(x)n c(x) = P(x)
y(x) = Q(x)Q(x)
n b 6= 0
n+ 1 b= 0 a6= 0
n+ 2 b= 0 a= 0
•c(x) = esxP(x)P(x)n
y(x) = esxQ(x)Q(x)
n s
n+ 1 s
n+ 2 s
•c(x) = c1(x) + c2(x)
y1(E1) : y00a(x)y0+b(x)y=c1(x)y2
(E2) : y00a(x)y0+b(x)y=c2(x)y1+y2
(E) : y00a(x)y0+b(x)y=c(x)