‚é™—pitul—tif de l— sé—n™e n¦Q Équ—tions différentielles

y0+ 2y= 4e1−2xE
fRf(x) = 4xe1−2xE
f0+ 2f= 4e1−2x
f0(x)=4e1−2x+ 4x(−2)e1−2x= 4e1−2x−8xe1−2x= 4e1−2x−2f(x)
f0(x) + 2f(x) = 4e1−2x−2f(x) + 2f(x) = 4e1−2xf
E
(E0)y0+ 2y= 0 y0=−2y
f0=af f(x) = Ceax C∈R
(E0)y(x) = Ce−2xC∈R
⇒gRE g −f(E0)
g E g0+ 2g= 4e1−2x
(g−f)0+ 2(g−f)=(g0+ 2g)−(f0+ 2f) = 4e1−2x−4e1−2x= 0
g−f(E0)
⇐g−f(E0)gRE
g−f(E0) (g−f)0+ 2(g−f) = 0
(g−f)0+ 2(g−f) = (g0+ 2g)−(f0+ 2f)=(g0+ 2g)−4e1−2x= 0 ⇒(g0+ 2g)=4e1−2x
g E
g E
g E g

g−f(E0) (E0)
y(x) = Ce−2xC∈Rg−f=Ce−2xE
g(x) = Ce−2x+f(x) = Ce−2x+ 4xe1−2x=e−2x(C+ 4xe)C∈R
E C
−2e0
g0(0) = −2e g0(0) = e−2·0(C+ 4 ·0·e) = C C =−2e
g0(x) = e−2x(−2e+ 4xe)=2e−2x+1(−1+2x)
(E)y00 −2y0+y=x2
f(x) = x2+ 4x+ 6 (E)
fR
•f0(x) = 2x+ 4
•f00(x) = 2
f00(x)−2f0(x) + f(x)=2−4x−8 + x2+ 4x+ 6 = x2
f(x) = x2+ 4x+ 6 (E)
2Rp(x) = c+bx+ax2
(a, b, c)∈R3p(E)p00 −2p0+p=x2
•p0(x) = 2ax +b
•p00(x) = 2a
p00 −2p0+p=x2⇔2a−4ax −2b+c+bx +ax2= (2a−2b+c)+(b−4a)x+ax2=x2
•ax2=x2a= 1
•x(b−4a) = 0 (b−4·1) = 0 b= 4
•(2a−2b+c) = 0 (2 ·1−2·4 + c)=0 c= 6
2 (E)f(x) = x2+4x+6
g(x) = (2x−5)ex+x2+ 4x+ 6 (E)
g(x) = r(x) + f(x)r(x) = (2x−5)ex
(r+f)00 −2(r+f)0+ (r+f)=(r00 −2r0+r)+(f00 −2f0+f)=(r00 −2r0+r) + x2
g(E)
(r+f)00 −2(r+f)0+ (r+f) = x2
r00 −2r0+r= 0 E0

•r(x) = (2x−5)ex
•r0(x) = 2ex+r(x)
•r00(x) = 2ex+r0(x)
r00 −2r0+r= 2ex+r0(x)−2r0(x) + r(x)=2ex−r0(x) + r(x) = 2ex−2ex−r(x) + r(x)=0
r E0g(E)
f0=af a ∈R
f(x) = Ceax C∈R
f0(x0) = Ceax0=y0C C =y0
eax0
C f0C
f0(x) = y0
eax0·eax
•2f0=f2f(−1) = 3
f0=1
2f f(x) = Ce 1
2x
C∈R
2f(−1) = 3 f(−1) = 3
2f(−1) = Ce−1
2Ce−1 2=3
2
C=3
2e−1
2
=3e1
2
2f0(x) =
3e1
2
2e1
2x=3
2e1
2(x+1) =3
2√e(x+1)
i(Eq)
Li0(t) + Ri(t) = E
uf0+vf =w(u, v, w)∈Rf0=af +b
(a, b)∈R
f0=af +b(a, b)∈R
f(x) = Ceax −b
a
C∈R
Li0(t) + Ri(t) = E f 0=af +b
i0(t) = −R
Li(t)+ E Li(t) = Ce−R
Lt−
E
L
−R
L
=Ce−R
Lt+E
R
C∈Rt= 0 i(0) = Ce−R
L·0+E
R= 0
C=−E
Ri(t) = −E Re−R
Lt+E
R=E R−e−R
Lt+ 1

i(t)t+∞
• − R
L<0 lim
t→+∞−R
Lt=−∞
•lim
t→+∞−R
Lt=−∞ lim
t→+∞
e−R
Lt= 0
•lim
t→+∞
e−R
Lt= 0 lim
t→+∞
e−R
Lt+ 1 = 1
•lim
t→+∞
e−R
Lt+ 1 = 1 lim
t→+∞
E
R−e−R
Lt+ 1=E
R
i(t)t+∞E
R