Chapitre 6 : Equations différentielles avec conditions initiales

T > 0f(t, y) : [0, T ]×R7→ R
tyt
y(t)∈C1([0, T ]) y0
(dy
dt (t) = f(t, y(t)) , t ∈[0, T ],
y(0) = y0.
0t
Rt
0
dy
dt (τ)dτ =y(t)−y(0)
y(t) = y0+Zt
0
f(τ, y(τ))dτ .
f(t, y)y
[0, t]

T > 0F(t, y) : [0, T ]×Rn7→ Rn
ty
y(t)∈C1([0, T ]) y0∈Rn
(dy
dt (t) = F(t, y(t)) , t ∈[0, T ],
y(0) = y0.
0t
y(t) = y0+Zt
0
F(τ, y(τ))dτ ;
0tv(t)=(vi(t))
iRt
0vi(τ)dτ

I(0) = I0
LR C
Q(0) = Q0
Q(t)
(Ld2Q
dt2+RdQ dt +1 CQ= 0
Q(0) = Q0,dQ
dt (0) = I0
I(t) = dQ
dt (t)
d
dt I
Q=−R
LI−1 CL Q
I
Q(0) = Q0,I(0) = I0.
n
n1
n= 2
(d2y
dt2=f(t , y,dy dt )
y(0) = y0,dy
dt (0) = y0
0
u=dy
dt
⇐====⇒
d
dt u
y=f(t , y,u) u
u(0)
y(0) =y0
0
y0

f(t, y)y
` t ∈[0, T ]y1
y2
|f(t, y1)−f(t, y2)| ≤ `|y1−y2|.
∂f
∂y (t, y)t∈[0, T ]
y
∂f
∂y (t, y)≤` ,
y
y
f
f(t, y) = 2 sin(y)
⇒`= 2
f(t, y)y
(dy
dt (t) = f(t, y(t)) , t ∈[0, T ],
y(0) = y0.
f(t, y)
y