Télécharger

TSI2
no2
(E)
x2y00 −x(2x2−1)y0−(2x2+ 1)y= 0 (E)
fR∗
∀x∈R∗, f(x) = 1
x
f0f00 f(E)
]0,+∞[
(E)
X
n>0
anxnR
x∈]−R, R[
y(x) =
+∞
X
n=0
anxn
y(E) ] −R, R[
x∈]−R, R[y0(x)y00 (x)
a0= 0 n2
an=2
n+ 1an−2.
a2ann
n n = 2p+ 1 p
p1a2p+1 a2p−1
p
a2p+1 =a1
(p+ 1)!.
X
p>0
x2p+1
(p+ 1)!
x7→ exp(x)x7→ exp(x2)
x g(x) =
+∞
X
p=0
x2p+1
(p+ 1)!
g(0)
x g(x) exp(x2)x
]0,+∞[ (E)
(E) ]0,+∞[f g

X
n>1
1
n;X
n>1
1
n√n;X
n>1
1
n2
α
X
n>1
sin2(nα)
n2
α]0, π[
f2πR
f(t)=1 t∈[−α, α],
f(t)=0 t∈]−π, −α[∪]α, π]
α=π
4
f[−2π, 2π]
bn(f)n∈N∗
a0(f)an(f)n∈N∗
+∞
X
n=1
sin2(nα)
n2=α(π−α) 2
+∞
X
n=1
sin2(n)
n2
α
+∞
X
n=1
1
n2
x(t)t
t7→ x(t)C1[0,+∞[
a > 0t>0x0(t) = ax(t)E1
x(0) = N
x(t)N a
t a = 0,2
a > 0m>N t >0x0(t) = ax(t)(m−x(t)) E2

TSI2
x0
0< x(0) < m
t>0x0(t)>0
x(0) > m
t>0z(t) = 1
x(t)
zC1[0,+∞[
E2
z0(t) + amz(t) = a(F2)
x(t) = m
1 + λ0e−amt
λ0m x(0)
y(t)
(x0(t) = ax(t)−bx(t)y(t)
y0(t) = −cy(t) + dx(t)y(t)
a, b, c, d
b= 0
x0(t) = ax(t)−bx(t)y(t)
d= 0
y0(t) = −cy(t)
. . .
x y C2
x(t) = c
d+u(t)
y(t) = a
b+v(t)
(u0(t) = −αv(t) + bu(t)v(t)
v0(t) = βu(t) + du(t)v(t)
α β a, b, c d
6= (0,0) u(t)v(t) 0
u(t)v(t)
(u0(t) = −αv(t)
v0(t) = βu(t)

u
u(t) = λsin(ωt +ϕ)ω=√αβ
v(t)α, λ, ω, ϕ t
x(t)
(x0(t) = x(t)(a−mx(t)) −bx(t)y(t)
y0(t) = −cy(t) + dx(t)y(t)
a, b, c, d
b= 0
−bx(t)y(t)
d= 0
y0(t) = −cy(t)
. . .
x y C2
x(t) = c
d+u(t)
y(t) = a
b+v(t)
(u0(t) = −αv(t) + bu(t)v(t)
v0(t) = βu(t) + du(t)v(t)
α β a, b, c, d k
6= (0,0) u(t)v(t) 0
u(t)v(t)
(u0(t) = −αv(t)
v0(t) = βu(t)
u
u(t) = λsin(ωt +ϕ)ω=√αβ
v(t)α, λ, ω, ϕ t