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]1 ; +∞[
y0−x
x2−1y= 2x
]−π/2 ; π/2[
y0(x)−tan(x)y+ (cos x)2= 0
y0−y= sin(2x)ex
y0+ 2xy = 2xe−x2
y0+ytan x= sin 2x]−π/2 ; π/2[
y0−(x+ 1)(y+ 1) = 0 y(0) = 1
(1 + x2)y0−(x+ 1)y= 2 y(0) = −1
(E): (1 −x)y0−y=g
g: ]−1 ; 1[ →R
g
g(x) =
+∞
X
n=0
bnxn
R≥1
(E)
y(x) =
+∞
X
n=0
anxn
R0≥1anbn
n∈N
a:R+→R
y0−a(t)y= 0
R+
h:R→C+∞
y0+y=h+∞
f:R→CC1f+f0−→
+∞`
f−→
+∞`
α
f:R→RC1f0+αf +∞
f+∞
α∈Cϕ:R→CT > 0
(E): y0+αy =ϕ(t)

yR(E)
t7→ y(t+T)
y(E)T
y(0) = y(T)
(E)T
α
R
(E): x2y0−y= 0
R
(ex−1)y0+ exy= 1
R
sh(x)y0−ch(x)y= 1
R
y0+y= max(x, 0)
xln xy0−(3 ln x+ 1)y= 0
R
xy0−y=x
xy0+y−1=0
xy0−2y=x4
x(1 + x2)y0−(x2−1)y+ 2x= 0
R
y0sin x−ycos x+ 1 = 0 (sin x)3y0= 2(cos x)y
(tan x)y0−y= 0 y(0) = 0
(tan x)y0−y= 0 y(0) = 1
R
x(x2−1)y0+ 2y=x2
α∈R R
xy0−αy = 0
α
f∈ C1(R+,R)gR∗
+
xy0−y=f(x)
g
f0(0)
fC2
gC2

(E)
(ln x)y0+y
x= 1
(E) ]0 ; 1[ ]1 ; +∞[
g]−1 ; +∞[\ {0}
g(x) = ln(1 + x)
x
g]−1 ; +∞[C∞
(E)C∞]0 ; +∞[
αR
E:xy0=αy
x7→ y(x)ER∗
+R∗ −
y(x)R∗
+R∗ −
C+C−y(x)
R∗
+R∗ −C+C−
y
α < 0α= 0 α > 0
α > 0C+C−
y
0< α < 1α= 1 α > 1
ER
α
α > 0I= ]0 ; +∞[
Eα:x2y00(x) + xy0(x)−α2y(x)=0
ϕ:y(x)7→ xy0(x)
y00 + 4y0+ 4y=e−2t
1 + t2
y00 +y= tan t
y00 +y= tan2t
]0 ; π[
y00 +y= cot x
y00 +y= cos(nt)
Pan
y00 +y=
+∞
X
n=0
ancos(nt)
f:R→RC2
f+f00 ≥0
∀x∈R, f(x) + f(x+π)≥0

f∈ C∞(R,C) 2π y ∈ C∞(R,C) 2π
y00 +y=f
f∈ C1(R+,R) +∞
y00 +y=f
R∗
+
y00 +y= 1/x
f(x) = Z+∞
0
e−tx dt
1 + t2
x > 0
Z+∞
0
sin t
tdt
y00 + 2xy0+ 2y= 0
]−1 ; 1[
4(1 −t2)y00(t)−4ty0(t) + y(t)=0
R
(1 + t2)y00(t)+4ty0(t)+2y(t)=0
(1 + t2)y00(t)+4ty0(t)+2y(t) = 1
1 + t2
h
xy00 +y0+y= 0
h(0) = 1
h]0 ; 2[
a, b:I→C(f1, f2)
E:y00 +a(t)y0(t) + b(t)y= 0
w:t7→ 
f1(t)f2(t)
f0
1(t)f0 2(t)
]0 ; +∞[
(E): ty00 + (1 −2t)y0+ (t−1)y= 0
ϕ(t)=et(E)
w(t)
(E)
(E)ϕ
(E)

q:R→R+
Ry00 +q(x)y= 0
f
f
−f f
∀x∈R, f(x)>0
f00
a∈Rf a
f a
f0(a) = 0
q:R→R+
Ry00 +q(x)y= 0
q[a;b]f
[a;b]
(E): y00(x) + q(x)y(x) = 0
f
q[0 ; +∞[ (E)
y00 +q(x)y= 0
f(E) [0 ; +∞[f0
+∞
f g f g
w=f0g−fg0
f g
q1, q2:I→Rq1≤q2
ϕ1ϕ2I
y00 +q1(x)y= 0 y00 +q2(x)y= 0
ϕ1
ϕ1x0∈I ϕ1
∃α > 0,∀x∈I∩[x0−α;x0+α], ϕ(x) = 0 =⇒x=x0
a < b ϕ1ϕ2[a;b]
ϕ1ϕ0
2−ϕ2ϕ0 1
ϕ
y00 + exy= 0
∀a∈R+,∃x∈[a;a+π], ϕ(x) = 0
(E): y00 + cos2(t)y= 0
u(E)u(0) = 1 u0(0) = 0
α, β
α < 0< β, u0(α)>0u0(β)<0
uR∗
−R∗ +
γ= max {t < 0|u(t) = 0}δ= min {t > 0|u(t)=0}