Divers.pdf

(E)xy′′ +2y′−xy =0R∗
+
R∑anxnn∈Na2n=1
(2n+1)!a2n+1=0
x∈] − R, R[f(x)=+∞
n=0
anxn
(bn)∑bnxnR′>0
g∶x+∞
n=0
bnxn(E)g(0)=1n∈N
bn=an
yR∗
+(E)z′
z∶xy(x)
f(x)(E)R∗
+
(E)2(x−x2)y′′(x)+(x−2)y′(x)−y(x)=0
y0∶xx−2
I]1,2[ ]2,+∞[
y(E)I z ∶xy(x)
x−2
ϕ∶x−2√x−1
x−2ϕ I ϕ′
(E)I4−3x2
2x(x−1)(x−2)=1 x−1 2(x−1)−2 x−2
(E) ]1,+∞[
(E) ]0,+∞[R
xy′′ −(x+3)y′+3y=0
y(0)=1
(S)∶x′=3x−2y
y′=2x−y

ln(n)xn
1+1
2+⋯ + 1 nxn
n(−1)nxn
ϕRϕ(x)=exp(exp(x)−1)
ϕ(x)=1+x+x2+5
6x3+o(x3)
ϕ(n)(0)n∈{0,1,2,3}
(Pn)P0=1n∈NPn+1=
n
k=0n
kPk
P1P2P3
n∈NPnn!
f(x)=+∞
n=0
Pn
n!xn
R f 0
x∈]−R, R[f′(x)=exf(x)
ϕ
(Hn)
exp tx −t2
2=+∞
n=0
Hn(x)tn.
f f(x, y)=x2+y2−2x−4y
n E =Rnu
E A Mn(R)ϕ E A
f E x =(x1, x2,...,xn)f(x)=x, ϕ(x)−2x, u
n=2A=3−1
−1 3 u=(5,1)
x∈R2f(x)=3x2
1+3x2 2−2x1x2−10x1−2x2
x0=(2,1)f
h=(h1, h2)f(x0+h)−f(x0)=ah2
1+bh2 2+ch1h2a, b, c
f x0
x E x, ϕ(x)>0
ϕ
ϕ f

f∶(x, y)x2y(x+y−4)∆={(x, y)∈R2, x 0, y 0, x +y6}
∆
f∆
(t, u)∈R2M(t, u)=(sin tcos u, cos tcos u, sin u)
S∶x2+y2+z2=1
S
S A =(1,0,1)
P∶x−y+z=3S∶x2+y2+z2=4S P
(x, y)∈R2∖{(0,0)} f(x, y)=xy x2+y2f(0,0)=0
fR2
fR2
fC1R2
∂f
∂x −y 2
∂f
∂y =0(x, y)=(u, ve−u/2)
a∈R
Rn[X]f
f(P)(X)=(X−a)P′(X)+P(X)−P(a).
f g E
dim( (f)∩(g))=(f)−(g○f)
KA B Mn(K)
dim( (AB)) dim( (A))+dim( (B))
R3[X]H={P∈R3[X], P (1)=0}
X H

F G
E n i ∈J0, nKΩi
(A, B)E(B)=i A ∪B=E
(Ωi)i∈J0, nKn=5i=3
P={z∈C,Im(z)>0}D={z∈C,z<1}f(z)=z−i
z+i
z∈P f(z)∈D f P D
f∶xe−1/xC∞R∗
+
n∈NPn∈R2n[X]x>0f(n)(x)=Pn1
xe−1/x
g(x)=f(x)x>0g(x)=0x0
gC∞R
g n
cos(1)
n n
(un)n∈N(vn)n∈N√unvn
n∈N∗xenx =1xn
(xn)n∈N∗
∑xn∑x2
n
xn+∞