ÉQUATIONS DIFFÉRENTIELLES LINÉAIRES

R
y0+y=−1
y0+y= ex
y0+ 2y=x2
y0+y=−1 + x2+ ex
(a, b)∈R2
Ry0−by = sin(ax)
y0+y
1 + x2=e−Arctan(x) x2]0,+∞[
y0−ytan(x) + cos2(x) = 0 ] −π
2,π 2[
y0+1
x−1y=1 x−1 (x−1)3]1,+∞[
2x(1 −x)y0+ (1 −x)y=1
√x]0,1[
R
y00 −5y0+ 4y= ex
y00 −4y0+ 4y= 2ex
y00 + 2y0+y= 4e−x
y00 + 4y= 3 cos2(x)
y00 + 2y0+ 2y= 2e−xsin(x)
y00 −4y0+ 3y= 2e3x+ cos(2x) + sin(2x)
y00 + 4y= cos(3x)
y(0) = 1
y0(0) = 0
]− ∞,0[ ]0,+∞[x2y0−y= 0
x2y0−y= 0 R
R(1 + x2)y00 + 2xy0= 0
(?) (1 + ex)y00 +y0−exy= 0.
fRg=f0+f
f(?)g
(?)

f: [0,1] −→ R
∀x∈[0,1], f0(x) + f(x) + Z1
0
f(t)dt = 0.
(E) (1 −x2)y00 −xy0+ 4y= Arccos(x) ] −1,1[
f(E)
g]0, π[g(t) = f(cos(t))
g g0g00 f
g(E0)
g(t)t∈]0, π[
x∈]−1,1[ f(x)
(E)
f]0,+∞[
∀x > 0, f0(x) = f1
x(?)
f]0,+∞[ (?)
f]0,+∞[f00 f0f
g g(t) = f(et)
Dgg
gDgg0g00
f
g
x > 0f(x)
f]0,+∞[ (?)
(" ]0,+∞[−→ R
x7−→ µ√x√3 cos √3
2ln x+ sin √3 2ln x #;µ∈R).
f:R−→ R
∀x∈R, f0(x) = f(−x).
f
f
y0+ 2xy = 1