SVE 101, TD Feuille 3 3. Equations di érentielles du second ordre

y(0) = 0 y0(0) = 1
y00 + 2y0−3y=−t+ 1
y00 + 2y0−3y=e2t
y00 + 2y0−3y=−t+1+e2t+ cos t
y00 −6y0+ 9y= 3 + et
y00 −3y0= 3 + t2
y00 +y=t+ sin(2t)
y00 + 5y0−6y=e2x(x2+ 1) (E).
yp=yp(x)yp(x) = Q(x)·e2x
x7→ Q(x)
y(0) = 0 y0(0) = −1
y00 +y0+y=x2ex
y00 + 2y0−3y=xsin x
y00 +y=xex
y00 +y=xcos x
y00 +y=x(cos x+ex)
y(0) = 1
y0(0) = 2
m ~a =1
m~
Fe~ Fe
~ O
~g =−g~ ~a =a(t)~
z(t)t a(t)
z(t)
m~g
z
z00 =−g.
z(0) = h

z
z00 =−g−k
mz0
t z0
M m
M ~
M y(t)M t
−−→
OM =y(t)~)
M
M y00 m
y0b≥0
y c > 0y
my00 +by0+cy = 0.
b= 0
b b2−4mc < 0
b b2−4mc > 0
y(t)t
b2−4mc < 0
Acos(ωx) + Bsin(ωx) = Csin(ωx +ϕ)C=√A2+B2ϕcos ϕ=B
√A2+B2
sin ϕ=A
√A2+B2
M λ
my00 +by0+cy =ksin λt
k
b > 0y(t)t