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f: [0 ; 1] →R
∀x∈[0 ; 1], f0(x) + f(x) + Z1
0
f(t) dt= 0
f:R→R
∀x∈R, f0(x) + f(−x)=ex
fR
∀x∈R, f0(x) = f(2 −x)
f:R→R
∀(x, y)∈R2, f(x+y)=exf(y)+eyf(x)

f
f y0+y+λ= 0
f(x) = Ce−x−λ.
Z1
0
f(t) dt=C(e −1)
e−λ
C(e −1)
e−λ=λ
λ=C(e −1)
2e
∀x∈[0 ; 1], f(x) = Ce−x−C(e −1)
2e
f
f0(x)=ex−f(−x)
f0
f00(x)=ex+f0(−x)=ex+ e−x−f(x)
f y00 +y= 2 ch x
f(x) = ch x+C1cos x+C2sin x
sh x−C1sin x+C2cos x+ ch x+C1cos x−C2sin x= ex
C1+C2= 0
f(x) = ch x+C(cos x−sin x)
f
f f
f00(x) = −f0(2 −x) = −f(x)
f y00 +y= 0
y(x) = λcos x+µsin x
−λ=λsin 2 −µcos 2
µ=λcos 2 + µsin 2
(1 + sin 2)λ= (cos 2)µ
λ= (cos 2)α µ = (1 + sin 2)α
f(x) = α(sin x+ cos(2 −x)) α∈R
f
x=y= 0 f(0) = 0
f(x+h)−f(x)
h=exf(h)+ehf(x)−f(x) h= exf(h)−f(0) h+eh−1 hf(x)
f(x+h)−f(x)
h−→
h→0exf0(0) + f(x)
f x f0(x) = f0(0)ex+f(x)
f
y0=y+Cexy(0) = 0
f(x) = Cxex