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n
F
dny
dtn, dn−1ydtn−1
, . . . , y , t
= 0
y ∈ Rm t
dnydtn
= f dn−1y
dtn−1, . . . , y , t
y(t)
I
∀t∈I
dn−1ydtn−1
, . . . , y , t
∈ Dm f
R
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n
Y (t) =
y
dydt
dn−2ydtn−2
dn−1ydtn−1
∈ Rm·n dY (t)dt
=
dydt
d
2
ydt2
dn−1ydtn−1
f dn−1ydtn−1
, . . . , y , t
n y(t)
Y (t)
y = f (t) y(t) =ˆ
f (t) dt
y = sin t y(t) =ˆ
sin t dt = − cos t + C C
Dm f ⊂ Rm·n+1
∆ = (yn−1, . . . , y1, y0, t0) ∈ Dm f.
y
t0 ∈ Dm y dn−1ydtn−1
(t0) = yn−1 . . . dy
dt(t0) = y y(t0) = y0
y = sin t (y0, t0)
y(t) = ´ sin t dt = −cos t + C y0 =
−cos t0 + C
⇒C = y0 + cos t0
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dy
dt =
−ry r > 0 y(0) = m > 0
y0(t) = m
y1(t) = y(0) +
tˆ 0
− ry0(u)
du = m − mrt
n
yn(t) = y(0) +
tˆ 0
− ryn−1(u)
du
yn+1(t) − yn(t) =m − r
tˆ 0
yn(u) du
−
m − r
tˆ 0
yn−1(u) du
= −rtˆ
0
yn(u) − yn−1(u) du
∀n0 yn+1(t) − yn(t) = m (−rt)n+1
(n + 1)!
y1(t) − y0(t) = −rmt yn(t) − yn−1(t) = m (−r)ntnn!
yn+1(t) − yn(t) = −rtˆ
0
yn(u) − yn−1(u)
du = −r
tˆ 0
m(−r)n tn
n! du
= m(−r)n+1tˆ
0
tn
n! du = m(−r)n+1 t
n+1
(n + 1)!
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yn(t) = [yn(t) − yn−1(t)] + [yn−1(t) − yn−2(t)] + . . . + [y1(t) − y0(t)] + y0(t)
= mn
k=1(−rt)k
k!
+ m = mn
k=0(−rt)k
k!
n→∞
−−−→me−rt
d
dt
me−rt
= m(−r)e−rt = −ry(t)
y = f (y, t) y ∈ Rm f
Rm y(t)
(y0, t0) t0 ∈ Dm y (y0, t0) ∈ Dm f
∀t∈Dm y (y(t), t) ∈ Dm f y(t) = y0 +tˆ
t0
f (y(u), u) du
y
y
y(t) = y0 +
tˆ t0
dydt (u) du = y0 +
tˆ t0
f (y(u), u) du
t
dy
dt =
d
dt
tˆ t0
f (y(u), u) du = f (y(t), t)
y(t0) = y0 +
t0
ˆ t0
f (y(u), u) du = y0
K f : K → R
∃m,M ∈K f (m) = inf {f (x) : x ∈ K } f (M ) = sup {f (x) : x ∈ K }
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K f : K → R
∀ε>0 ∃δ>0 ∀x1,x2∈Dm f x1 − x2 < δ ⇒ f (x1) − f (x2) < ε.
δ 0 > 0
δ f : (0, δ 0) →R
+
limx→0+
δ f (x) = 0 0 < x1 − x2 < δ 0 f (x1) − f (x2) δ f
x1 − x2
δ f
δ f (x) = sup {f (x1) − f (x2) : 0 < x1 − x2 < x}.
δ 0 ε = 1
f L
∀x1,x2∈Dm f f (x1) − f (x2) L · x1 − x2
δ f (x) = Lx
f : R ⊃ I → Rm
∀t0
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= f (t) −
mk=1
t́ t0
f k(u) du · f k(t)
t́ t0
f (u) du
= f (t) − t´ t0
f (u) du, f (t)t́ t0
f (u) du
f (t) −
t́ t0
f (u) du
· f (t)t́ t0
f (u) du
= 0
t = t0
y = f (y, t) y(t0) = y0
f B(y0, b) × [t0 − a, t0 + a] a,b > 0 M f (y, t) y(t) [t0 − α, t0 + α] α = min
a, b
M
B(y0, b)
∆t > 0
[u]∆t =
t0 + ∆tu−t0∆t
, u t0
t0 − ∆tt0−u∆t
, u < t0
[·]
[u]∆t
tt0t0 − ∆t
t0 + ∆tt0 − 2∆tt0 + 2∆t
t0 − ∆t
t0 + ∆t
[u]∆t
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∀u∈R |u − [u]∆t| t0 t < t0
y(∆t)(t) = y0 +
t0+k∆tˆ t0
f
y(∆t)([u]∆t), [u]∆t
du +
tˆ
t0+k∆t
f
y(∆t)([u]∆t), [u]∆t
du
= y(∆t)(t0 + k∆t) +
tˆ
t0+k∆t
f
y(∆t)([u]∆t), [u]∆t
du
y(∆t)(t) y(∆t)(t0+k∆t)
u ∈ (t0+k∆t, t0+(k +1)∆t)
u
[u]∆t = t0 + k∆t
y(∆t)(t) = y(∆t)(t0 + k∆t) + (t − t0 − k∆t)f
y(∆t)(t0 + k∆t), t0 + k∆t
y(∆t)(t0 + k∆t) − y0 Mk∆t
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|t − t0| α k∆t α Mk∆t Mα b y(∆t)(t0 + k∆t), t0 + k∆t
f y(∆t)(t)
y(∆t)(t) − y0 y(∆t)(t0 + k∆t) − y0 + |t − t0 − k∆t| · M k∆tM + |t − t0 − k∆t| · M = |t − t0| · M
y(∆t)(t0 + k∆t) − y0 t0 + k∆t < t t0 + (k + 1)∆t
y(∆t) M
y(∆t)(t1) − y(∆t)(t2) =
t1ˆ t2
f
y(∆t)([u]∆t), [u]∆t
du
t1ˆ
t2
M du
= M |t1 − t2|
y : [t0 − α, t0 + α] → B(y0, b)
Φ(y) = sup
y(t) − y0 −
tˆ t0
f (y(u), u) du
: t ∈ [t0 − α, t0 + α].
Φ(y) = 0 ⇔ y(t) = y0 +tˆ
t0
f (y(u), u) du
Φ
yn ⇒ y [t0 − α, t0 + α] ⇒ Φ(yn) → Φ(y).
f B(y0, b) × [t0 − α, t0 + α] δ f
∀ε>0 ∃N 0 ∀t∈[t0−α,t0+α] ∀nN 0 yn(t) − y(t) < η,
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η > 0 δ f (η) < ε2α
ε > 0
yn(t) − y(t) < ε2
|Φ(yn) − Φ(y)|
=
suptyn(t) − y0 −
tˆ t0
f (yn(u), u) du
− supty(t) − y0 −
tˆ t0
f (y(u), u) du
supt
yn(t) − y0 −
tˆ t0
f (yn(u), u) du
−y(t) − y0 −
tˆ t0
f (y(u), u) du
supt
yn(t) − y(t) +tˆ
t0
(f (y(u), u) − f (yn(u), u)) du sup
tyn(t) − y(t) + sup
t
tˆ
t0
(f (y(u), u) − f (yn(u), u)) du<
ε
2 + sup
t
tˆ
t0
f (y(u), u) − f (yn(u), u) du
ε2
+t
tˆ
t0
δ f (η) du
< ε
2 + sup
t
tˆ
t0
ε
2α du
ε
2 +
ε
2α sup
t|t − t0| = ε
2 +
ε
2αα = ε
y(∆t)(t) lim∆t→0+
Φ
y(∆t)
= 0
u ∈ [t0 − α, t0 + α] y(∆t)([u]∆t, [u]∆t − y(∆t)(u), u = y(∆t)([u]∆t) − y(∆t)(u)2 + |[u]∆t − u|2
<√
M 2 + 1∆t
|[u]∆t
−u
|
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tˆ t0
f y(∆t)([u]∆t), [u]∆t − f y(∆t)(u), u du
tˆ
t0
δ f √
M 2 + 1∆t
du
|t − t0| δ f
√ M 2 + 1∆t
δ f f
0 ∆t → 0
Φ
y(∆t)
= sup
y
(∆t)(t) − y0 −tˆ
t0
f
y(∆t)(u), u
du
: t ∈ [t0 − α, t0 + α]
|t − t0| δ f √
M 2 + 1∆t
αδ f √
M 2 + 1∆t ∆t→0−−−→ 0
F
∀ε>0 ∃δ>0 ∀f ∈F ∀x,y x − y < δ ⇒ f (x) − f (y) < ε.
y(∆t) δ f (x) = M x
(∆t)k → 0 y(∆t)k ⇒ y
Φ(y) = limk→∞
Φ y(∆t)k = 0
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y(t)
R t ∈ [t0 − a, t0 + a] t0 ∈ R a > 0
∀t∈[t0−a,t0+a] y(t) K + L t
ˆ t0
y(u) du K, L 0
∀t∈[t0−a,t0+a] y(t) K exp (L |t − t0|)
L = 0
L > 0 K = 0 K = 1n
n ∈ N
∀n∈N y(t) 1n
exp L |t − t0| ⇒ y(t) inf
1
n exp L |t − t0|, n ∈ N
= 0.
K > 0
y(t)
K + L
t́ t0
y(u) du
1 ⇒ y(t)
K L
+
t́ t0
y(u) du
L.
K L
+
tˆ t0
y(u) du = v(t) dvdt =
y(t)
t t0 −y(t) t < t0
tˆ
t0
y(u)
K L
+
ú t0
y(w) dw
du
tˆ t0
L du
sgn(t − t0)t
ˆ t0
y(u)
v(u) dv L |t − t0|ln v(t)v(t0) L |t − t0|
ln
K L
+
t́ t0
y(u) du
K L
L |t − t0|
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K L
+
t́ t0
y(u) du
K L
exp L |t − t0|
y(t) K + L
tˆ t0
y(u) du
K exp L |t − t0|,
y = f (y, t)
f : B(y0, b) × [t0 − a, t0 + a] → Rm y
∃L>0 ∀t∈[t0−a,t0+a] ∀y1,y2∈B(y0,b) f (y1, t) − f (y2, t) L y1 − y2 .
y1 y2 I ⊂ [t0 − a, t0 + a] yk(t0) = y0 k = 1, 2 t ∈ I
y1 y2 I k = 1, 2 t ∈ I
yk(t) = y0 +
tˆ t0
f (yk(u), u) du
y2(t) − y1(t) =
tˆ t0
(f (y2(u), u) − f (y1(u), u)) du
tˆ t0
f (y2(u), u) − f (y1(u), u) du
t
ˆ t0
L y2(u) − y1(u) du = L t
ˆ t0
y2(u) − y1(u) du . K = 0 L = L y(t) = y2(u) − y1(u) y1(t) = y2(t)
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¨
dy
dt = f (y, t) y(t0) = y0
f : Rm+1
→ Rm
B(y0, b) × [t0 − a, t0 + a] = Q
a,b > 0 sup(y,t)∈Q
f (y, t) = M y Q
∃L>0 ∀t∈[t0−a,t0+a] ∀y1,y2∈B(y0,b) f (y1, t) − f (y2, t) L y1 − y2 .
|t − t0| α α = min
a, b
M , 1L
T : X → X (X, d) T
a ∈ X T (a), T (T (a)), . . . T
T c 0
n0 ∞
i=n0
d(ai, ai+1) < ε d(an, am) < ε
m, n n0 (an) (X, d)
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an → p p ∈ X T T (an) → T ( p) an = T (an−1) → T ( p) p = T ( p)
T (q ) = q d( p, q ) = d(T ( p), T (q ))
c·
d( p, q )
d( p, q ) = 0 c
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sup|t−t0|α
F (y1)(t) − F (y2)(t) = sup|t−t0|α
tˆ t0
f (y1(s), s) ds −tˆ
t0
f (y2(s), s) ds
= sup|t−t0|α
tˆ t0
f (y1(s), s) − f (y2(s), s) ds sup|t−t0|α
tˆ t0
L · y1(s) − y2(s) ds
sup|t−t0|α
L ·tˆ
t0
sup|s−t0|α
y1(s) − y2(s) ds sup|t−t0|α
L y1(t) − y2(t) sup|t−t0|α
tˆ t0
ds
Lα sup|t−t0|α
y1(t) − y2(t)
Lα < 1 F
F yn+1(t) = F (yn)(t)
y0(t) = y0 E
W ⊂ Rm × R1 × Rl f : W → Rm y W
Γ f y
∀(y1,t,λ), (y2,t,λ)∈Γ f (y1, t , λ) − f (y2, t , λ) LW y1 − y2 .
LW W
M M = sup {Mh : h = 1} M h Mh =
M · hh · h = h · M · hh h · M f : W
→Rn W
⊂Rm C 1
sup {Df (y) : y ∈ W } = L
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W y(t) = y1 + th h = y2 − y1 y(0) = y1 y(1) = y2
d(f ◦ y)dt
= Df (y(t)) · dydt
= Df (y(t)) · h.
f (y2) − f (y1) = (f ◦ y)(1) − (f ◦ y)(0) =
1ˆ 0
df ◦ ydt
dt
1ˆ
0
df ◦ ydtdt
=
1ˆ 0
Df (y(t)) · h dt 1ˆ
0
L y2 − y1 dt = L · y2 − y1
f : W → Rm C 1 y y t λ
W f V
V ⊂ W V
Dyf W
f : U → Rm (y0, t0) ∈ U I t0 I yk(t0) = y0 k = 1, 2
a b B(y0, b) × [t0 − a, t0 + a] f y
y = f (y, t) f : U → Rm U ⊂ Rm×R U (y0, t0) ∈ U y1 y2 yk(t0) = y0 k = 1, 2 y1 ≡ y2
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I
α = inf {t ∈ I : t t0 ∧ y1(t) = y2(t)}
α∈
I
y1(α) = y2(α) y1 y2
[t0, α)
(y1(α), α) α y1 ≡ y2 α y1 ≡ y2 α
t0
y(t)
a
(y(a), a) ∈ U α > 0 y : [a − α, a + α] → Rm y(a) = y(a)
y = y [a
−α, a] ỹ = y
∪y
y
(a, a+α) ỹ t1
y y
f y
U (y0, t0) ∈ U y(t) y(t0) = y0
(yα) α ∈ A yα(t0) = y0
y =α∈A
yα
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yβ
I β I β β ∈
A
I β y
yγ γ ∈ A y ≡ yγ Dm y ⊃ Dm yγ yγ y ≡ yγ
K ⊂ U ⊂ Rm K U ε > 0
L ⊂ U ∀x∈K B(x, ε) ⊂ L
f : U → Rm U ⊂ Rm × R f y y(t)
(a, b)
Y (t) : (a, b) → U, Y (t) = (y(t), t).
K
⊂U Y −1(K )
K
0t Y −1(K ) Y −1(K ) (a, b)
Y a b
Y −1(K ) = (a, b)
Y −1(K ) {a} {b}
K
∃η>0 ∃α>0 ∀(y0,t0)∈K B(y0, η) × (t0 − α, t0 + α) ⊂ L ⊂ U
L f L M
y(t) y(t0) = y0
[t0−α, t0+α] α = min {α, ηM } t0 ∈ Y −1(K ) |a − t0| |b − t0|
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α y ∪ y y
dy
dt = A(t)y + B(t)
I A(t) B(t)
y
y(t) = y0 +
tˆ t0
A(τ )y(τ ) dτ +
tˆ t0
B(τ ) dτ
y(t) y0 +
tˆ t0
A(τ )y(τ ) dτ +
tˆ
t0
B(τ ) dτ .
t
∈ I τ t t0
A(τ )
B(τ )
β = sup B(τ ) α = sup A(τ ) τ t t0
y(t) y0 + β |t − t0| + αtˆ
t0
y(τ ) dτ.
y(t)
y(t)
(
y0
+ β
|t−
t0|)eα|t−t0|
I
I
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0y
(x, y)
x0 = x1 = 1 xn+2 = xn+1 + xn n 0
n > 1 xn > (1, 6)n−1
1 y = x2
x(t) 0y
g
v2/2 gy
x dxdt
x x(0) = 2
dxdt (t̂)
t̂ x(t̂) = −1
α > 0
α
x(t) r
c
dx
dt = rx − cxt.
t = 0 x(0) = 1
t̂ x(t̂)
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h1 = 1500
h2 = 500 50 /
0xy y = c1 exp(c2x)
dx
dt − x ctg t = 0 x
π2
= 1
(0, π)
dydt = ry 0 < r 0
y = t(1 − cos
|y|)
y = P (y, t) P
R
y = y2
y+t y(0) = 2
0 y(10) 2e10
x
= y
y = tg x + 3√
t
x(0) = 0 y(0) = 1 [0, +∞) y(t) > 12 t 0
(x0, y0) ∈ R2 x
= x2 + y2 x(0) = x0
y = 2xy2 y(0) = y0
(0, x0, y0)
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y = f (x, y)
{(x, y) : y > 0} f C 1 x
f : R → R [f (a) − f (b)](a − b) 0 a, b ∈ R y1 y2
y = f (y), y(t0) = y0,
t t0 y1(t) = y2(t)
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dy
dt = f (y,t,λ)
f : G →Rm
G = U × B(λ0, c) ⊂Rm+1
×Rl
.
(y0, t0) ∈ U y(t, λ) y(t0, λ) = y0 λ
[b, a]
t0 ∈ [b, a] y(t, λ0) [b, a]
dy
dt = f (y,t,λ)
f : T D × B(λ0, c) →Rm
T D × B(λ0, c) ⊂Rm+1
×R
c,D > 0
T D = {(y, t) : t ∈ [b, a], y − y(t, λ0) D}
y(t, λ0) λ = λ0
y(t0, λ0) = y0
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∀λ∈B(λ0,c) f (y,t,λ) (y, t) y
∃L>0 ∀(y1,t),(y2,t)∈T f (y1, t , λ0) − f (y2, t , λ0) L y1 − y2
∀ε>0
∃δ>0
∀(y,t)
∈T
λ
−λ0
< δ
⇒ f (y,t,λ)
−f (y,t,λ0)
< ε
c1 > 0 λ λ − λ0 < c1 y(t, λ) (y0, t0) [b, a]
λn → λ0 y(t, λn)⇒ y(t, λ0) [b, a]
α(λ) λ ∈ B(λ0, c)
α(λ) = inf {t ∈ [t0, a] : y(t, λ) − y(t, λ0) D}
α(λ) = a
y(t, λ) [t0, α(λ)]
y(t, λ) β ∈ (t0, α(λ)) (y(t, λ), t) ∈ T t β α(λ) < β < α(λ)
λ ∈ B(λ0, c)
∀(y,t)∈T f (y,t,λ0) − f (y,t,λ) < ε
∀t∈[t0,α(λ)] y(t, λ) − y(t, λ0) ε · (a − t0)eL(a−t0).
y(t, λ) − y(t, λ0) =
tˆ t0
[f (y(u, λ),u ,λ) − f (y(u, λ0),u ,λ0)] du
tˆ
t0
f (y(u, λ),u ,λ) − f (y(u, λ),u ,λ0) du
+
tˆ t0
f (y(u, λ),u ,λ0) − f (y(u, λ0), u , λ0) du
ε(t − t0) + Ltˆ
t0
y(u, λ) − y(u, λ0) du ε(a − t0) + Ltˆ
t0
y(u, λ) − y(u, λ0) du.
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ε ε[(a − t0)eL(a−t0)] D2 c1 > 0 λ − λ0 < c1 ∀(y,t)∈T f (y,t,λ) − f (y,t,λ0) < ε
∀λ∈B(λ0,c) ∀t∈[t0,α(λ)] y(t, λ) − y(t, λ0) D
2 ,
α(λ) α(λ) = a y(t, λ)
t ∈ [t0, a] λn → λ0 εn → 0 n ∈ N
∀(y,t)∈T f (y,t,λ0) − f (y,t,λn) < εn.
supt∈[t0,a]
y(t, λ0) − y(t, λn) εn[(a − t0)eL(a−t0)] → 0.
b = t0
b < t0 t0 = a
f C 1 y
f : U → Rm dydt
= f (y, t) f C 1 y
(y0, t0) ∈ U (ỹ, t̃) ∈ B((y0, t0), c) ⊂ U y(t, ỹ, t̃) y(t, y0, t0) [b, a]
(ỹ, t̃) (y0, t0) [b, a]
y(t, ỹ, t̃)⇒ y(t, y0, t0) (ỹ, t̃)
→(y0, t0)
λ = (ỹ, t̃) F (Y , T , λ) = f (Y + ỹ, T + t̃)
F [U − (y0, t0)] × B((y0, t0), c) c = dist {(y0, t0), ∂U } U −(y0, t0) U (y0, t0) F C 1 Y
dY (T, λ)
dT = F (Y , T , λ), Y (0, λ) = 0
8/17/2019 Skrypt mat
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Y (t, λ)
y(t, ỹ, t̃) = Y (t − t̃, λ) + ỹ
y(t̃, ỹ, t̃) = Y (t̃ − t̃, λ) + ỹ = Y (0, λ) + ỹ = ỹ.
y(t, ỹ, t̃)
dy(t, ỹ, t̃)
dt =
dY (t − t̃, λ)dt
.
T = t − t̃ dY (T, λ)
dt =
dY (T, λ)
dT ·
dT
dt = F (Y , T , λ) = f (Y + ỹ, T + t̃)
= f (Y (T, λ) + ỹ, T + t̃) = f (y(t, ỹ, t̃), t).
λ
dy
dt = λ2 + y2, y(0) = 1, λ0 = 0.
(−∞, 1) y(t, 0) = 11−t λ > 0 ∃α λ = α2
yˆ 1
1
α2 + u2 du = t.
y = α · tgtα + arctg 1α .
limα→0 α · tg
tα + arctg
1
α
=
1
1 − t
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t 0
[0, ∞) ∩ Dm y(t, λ) =
t 0 : tα + arctg 1α
< π
2
.
t∗ t∗α + arctg 1α
= π2
tg(t∗α) · tg
arctg 1
α
= 1
tg(t∗α) = α
t∗ = arctg α
α K > −∞ [−K, t∗]
λ0 ∈ (λ0 − c, λ0 + c) c > 0 dy
dt = f (y,t,λ) f : U × (λ0 − c, λ0 + c) → Rm, U ⊂ Rm+1,
f C 1 y λ
y(t, λ0) (y0, t0) ∈ U I
λ λ0 y(t, λ)
I
t ∈ I
z (t, λ0) = ∂y(t, λ)
∂λ
λ=λ0
t λ0
∂z
∂t(t, λ0) = Dyf (y,t,λ0)
y=y(t,λ0)
· z (t, λ0) + ∂ f ∂λ
(y(t, λ0), t , λ)λ=λ0
z (t0, λ0) = 0
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ε0 > 0
T ε0 = {(y, t) : t ∈ I, y − y(t, λ0) ε0}
U η > 0
|λ
−λ0
|< η
y(t, λ) t ∈ I
y(t, λ), t ∈ T ε0
t ∈ I
w(t, λ) = y(t, λ) − y(t, λ0)
λ − λ0 0 < |λ − λ0| < η
w(t, λ) =
t
´ t0 f
y(τ, λ), τ , λ
− f
y(τ, λ0), τ , λ0
dτ
λ − λ0
=
tˆ t0
f
y(τ, λ), τ , λ
− f
y(τ, λ), τ , λ0
λ − λ0 dτ
+
tˆ t0
f
y(τ, λ), τ , λ0
− f
y(τ, λ0), τ , λ0
λ − λ0 dτ
g1(w,τ ,λ) = f y(τ, λ0) + (λ − λ0)w,τ ,λ− f y(τ, λ0) + (λ − λ0)w,τ ,λ0
λ − λ0
g2(w,τ ,λ) =f
y(τ, λ0) + (λ − λ0)w,τ ,λ
− f
y(τ, λ0), τ , λ0
λ − λ0− Dyf (y,τ ,λ0)
y=y(τ,λ0)
· w.
w < ε0λ−λ0
w(t, λ) =
tˆ t0
g1(w(τ, λ), τ , λ) dτ +
tˆ t0
g2(w(τ, λ), τ , λ) dτ
+
tˆ t0
Dyf (y,τ ,λ0)y=y(τ,λ0)
w(τ, λ) dτ
8/17/2019 Skrypt mat
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g1 λ = λ0
g1(w,τ ,λ0) = ∂f
∂λ(y(τ, λ0), τ , λ)
λ=λ0
.
w
∀R,ε>0 ∃δ1(ε,R)>0 ∀τ ∈I ∀0
8/17/2019 Skrypt mat
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|λ − λ0| < δ0(ε)√ R2+1 j
∂f
∂λ(y(τ, λ0) + (λ − λ0)w,τ, λ̄)|λ=λj
[ j] −
∂f
∂λ(y(τ, λ0) + (λ − λ0)w,τ, λ̄)|λ=λ0
[ j]
<
ε
√ m,
g1(w,τ ,λ) − g1(w,τ ,λ0) < ε.
δ 1(ε, R) = minε0R
, δ0(ε)√ R2+1
g2(w,τ ,λ)
∀R,ε>0 ∃δ2(ε,R)>0 ∀τ ∈I ∀0
8/17/2019 Skrypt mat
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g2(w,τ ,λ) = g2(w,τ ,λ) − g2(0, τ , λ) sup {Dwg2(w,τ ,λ) : w R} · w < ε.
δ 2(ε, R) = minε0R
, δ 0(ε, R)
η1 = min {δ 1(1, R), δ 2(1, R)}
∃M 0
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∀t∈I λ w(t, λ) (M + 2) |I | exp{L|I |}.
I λ
= I I λ
w(t, λ) I λ = I M 0 = R η1 = δ min
w(t) := w(t, λ)
dw
dt = g1(w,t,λ) + g2(w,t,λ) + Dyf (y,t,λ0)
y=y(t,λ0)
· w
w(t0) = 0 g1 λ = λ0
g2 g(w,t,λ0)≡
0
λ = λ0 g1 g2 f w Dyf (y,t,λ0)
y=y(t,λ0)
C 1 f
λ = λ0 g1 g2 w g1 t
C 1 f
w λ = λ0 g1
g2 w L
λ → λ0 λ g1 g2
δ 1(ε, R) δ 2(ε, R)
R = M 0 |λ − λ0| < η1
δ = min
η1, δ 1
ε
2, M 0
, δ 2
ε
2, M 0
.
w(t, λ) I
λ → λ0 dw
dt = g1(w,t,λ0) + Dy(y,t,λ0)
y=y(t,λ0)
· w
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w(t0) = 0 I
I
limλ→λ0 w(t, λ) = ∂y(t, λ)∂λ ,
z (t, λ0)
z (t, λ0) λ0
λ0
y(t, λ0) λ0
λ→
λ0 z (t, λ)⇒ z (t, λ0)
t z (t, λ0) (t, λ0)
λ λ = (λ(1), . . . , λ())
∂y(t,λ)
∂λ(j) |λ=λ0 j = 1, . . . ,
Dλy(t, λ)|λ=λ0
∂
∂tDλy(t, λ)|λ=λ0 = Dyf (y,t,λ0)|y=y(t,λ0)Dλy(t, λ)|λ=λ0 + Dλf (y(t, λ0), t , λ)|λ=λ0.
λ0 = (ỹ0, t̃0)
λ = (y0, t0) T = t − t0 Y = y − y0 dY
dT = F (Y , T , λ) = f (Y + y0, T + t0), Y (0) = 0
y(t, λ) = Y (t − t0, λ) + y0 y(t0, (y0, t0)) = y0
y0 t
∂y(t, (y0, t0))
∂y0|λ=λ0 =
∂
∂y0[Y (t − t0, (y0, t0)) + y0]|λ=λ0 =
∂
∂y0Y (t − t0, (y0, t0))|λ=λ0 + I.
8/17/2019 Skrypt mat
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z (t, λ0) z
∂z (t, λ0)
∂t = DY F (Y , T , λ)|Y =Y (t−t̃0,λ0) · z (t, λ0) +
∂
∂λF (Y , T , λ)|λ=λ0
= DY f (Y + y0, T + t0)|Y =y(t,λ0)−ỹ0 · z (t, λ0) + ∂ ∂λ f (Y + y0, T + t0)|λ=λ0= Dyf (y, T + t̃0)|y=y(t,λ0) · z (t, λ0) +
∂
∂λf (y, t)|λ=λ0.
z (t, λ0) = ∂y(t, λ)
∂y0|λ=λ0 − I
∂ ∂t∂y(t, λ)∂y0 |λ=λ0 − I = ∂ ∂t ∂y(t, λ)∂y0 |λ=λ0
= Dyf (y, t)|y=y(t,λ0)
∂y(t, λ)
∂y0|λ=λ0 − I
+ Dyf (y, t)|y=y(t,λ0)
= Dyf (y, t)|y=y(t,λ0) · ∂ y(t, λ)
∂y0|λ=λ0
∂y(t,λ)∂y0
|λ=λ0 = I z (t0, λ0) = 0
∂
∂t
∂y(t, (y0, t0))
∂t0 |λ=λ0 = Dyf (y, t)
|y=y(t,λ0)
∂y(t̃0,(y0,t0))∂t0
|λ=λ0 = −f (ỹ0, t̃0)
∂x∂µ
µ=0
x = −x2t + 2µ x(1) = 1 x = x + t sin(xt) x(1) = µ x = x + sin(xt) x(−1) = µ x = 2t + µx2 x(0) = µ − 1
x − x = (x + 1)2 − (µ + 1)x2 x(0) = 12
x(0) = −1
y = −1t − 1
t2 + ty2 ∂y
∂t0
t0=1
y(t0) = 1
8/17/2019 Skrypt mat
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y = x3 + 2
xy − y
2
x
(1, µ) µ = −1
dxdt = x + µy
2
dydt = x + y
x(0) = 1 y(0) = 0 µ = 0
8/17/2019 Skrypt mat
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8/17/2019 Skrypt mat
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f C 1 G dydt
= f (y)
ϕt(x) t ∈ R G
ϕt(x) = y(t, (x, 0)).
y(t, (x, 0))
y(0) = x t ∈ R y(t)
ϕt ◦ ϕs = ϕt+s.
dϕt
dt =
dy(t, (x, 0))
dt = f (y(t, (x, 0))) = f (ϕt(x))
dϕt ◦ ϕsdt
= f (ϕt ◦ ϕs)dϕt+s
dt =
d(t + s)
dt
dϕt+s
d(t + s) = 1 · f (ϕt+s)
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ϕt ◦ ϕs|t=0 = ϕ0 ◦ ϕs = ϕs
ϕt+s|t=0 = ϕs.
ϕt
◦ϕs ϕt+s dy/dt = f (y)
t ϕt(x)
C 1 x Dxϕt(x)
d
dtDxϕ
t(x) = Dyf (y)|y=ϕt(x) · Dxϕt(x)
Dxϕ0(x) = I
y = y2 + 1
y(0) = x
arctg y(t)
−arctg x = t
y(t) = tg(t + arctg x) = ϕt(x).
ϕt ◦ ϕs = tg(s + arctg ϕtx) = tg(s + arctg (tg(t + arctg x))) = tg(s + t + arctg x)
ϕt+s
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y = 1 + y + y2
y = ay + by
y
y
= φt
y0
y0
= exp
t 0 1
b a
y0
y0
R2
y =
0 1
−1 0
y +
0
−1
y = f (y) Rm divf = 0
m φt
E Vol(φt(E )) = Vol(E )
φt(x) = tg(t + x)?
y = (1 + y) sin(sin y)
ddx
φt(x)|x=0
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dy
dt = A(t) · y + B(t)
Rm
A(t) m × m I B(t) m
I y(t) m
an(t)y(n) + an−1y(n−1) + . . . + a1y + a0y = b(t)
n R ai(t) an(t) ≡ 0
dy
dt = A(t)y + B(t)
I A(t) B(t)
V y = Ay V B
y = Ay + B
8/17/2019 Skrypt mat
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V C (I )
V B V C (I ) V + v v ∈ C (I ) y1 ∈ V B y2 ∈ V B y1 − y2 ∈ V
y1, y2 ∈ V y = y1 + y2 dy
dt =
d(y1 + y2)
dt =
dy1dt
+ dy2
dt = Ay1 + Ay2 = A(y1 + y2) = Ay
day
dt = a
dy
dt = aAy = A(ay)
V
C (I )
y1
∈ V B
dydt
= Ay1 + B
y2 − y1 ∈ V dy2dt
= d(y2 − y1)
dt +
dy1dt
= A(y2 − y1) + Ay1 + B = Ay2 + B
y2 ∈ V B y2 ∈ V B
d(y2 − y1)dt
= dy2
dt − dy1
dt = Ay2 + B − Ay1 − B = A(y2 − y1)
y2 − y1 ∈ V
{y1, . . . , yk}
{y1, . . . , yk} V t ∈ I {y1(t), . . . , yk(t)} Rm t ∈ I {y1(t), . . . , yk(t)} Rm
⇒ {y1, . . . , yk}
k j=1
α jy j ≡ 0 ⇒ α1, . . . , αk = 0.
8/17/2019 Skrypt mat
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t0 ∈ I (α1, . . . , αk) = 0 k
j=1
α jy j(t0) = 0.
y(t) =k
j=1α jy j
I
y(t0) = 0 y(t) = 0
y(t) ≡ 0 ⇔ k j=1
α jy j ≡ 0 (α1, . . . , αk) = 0
⇒ ⇒ (α1, . . . , αk) = 0
k j=1
α jy j ≡ 0.
∀t∈I k
j=1α jy j(t) = 0
V
{y1, . . . , yk
} ⊂V
{y1, . . . , yk} k = m
⇒ k m Rm k < m t0 ∈ I {y1(t0), . . . , yk(t0)} x ∈ Rm {y1(t0), . . . , yk(t0), x(t0)}
yk+1
yk+1(t0) = x(t0)
{y1, . . . , yk+1} {y1, . . . , yk}
⇐ {y1, . . . , ym} ym+1 {y1, . . . , ym+1}
8/17/2019 Skrypt mat
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t ∈ I {y1(t), . . . , ym+1(t)} Rm
dim V = m
I
M (t)
M (t) ∈ M m×m
M (t) I C 1
t0
∈I M (t0)
dM dt
= AM
⇒ M
⇐ dM dt
= AM M
m
M
P m
M (t)P
d(M (t)P )
dt =
dM (t)
dt · P = AM (t)P = A(M (t)P ).
U m×m M (t)U U M (t)
dU M (t)
dt = U
dM (t)
dt = U AM (t) = A(U M (t))
M (t, t0) = M (t)M (t0)−1
M (t0, t0) = I
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M (t) y0 Rm
M (t, t0)y0
y(t0) = y0
y(t) = M (t, t0)y0
y(t) = M (t)C C ∈ Rm
P = y0
M (t0, t0) = I
C = M (t0)−1y0
y1, . . . , ym ∈ Rm
W (y1, . . . , ym)(t) = det(y1(t), . . . , ym(t)).
y1, . . . , ym
C m−1
W (y1, . . . , ym)(t) = det
y1(t) . . . ym(t)
y1(t) . . . ym(t)
y(m−1)1 (t) . . . y
(m−1)m (t)
.
W (t)
y1, . . . , ym
∃t0∈I W (y1, . . . , ym)(t0) = 0 ∀t∈I W (y1, . . . , ym)(t) = 0
8/17/2019 Skrypt mat
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y1(t), . . . , ym(t)
dydt
= Ay
dW (t)
dt = trA(t) · W (t).
M = [xij]
det M =m
j=1
(−1)i0+ jxi0 jM i0 j =mi=1
(−1)i+ j0xij0M ij0
i0, j0 = 1, . . . , m M ij
i j M
∂ det M
∂xij= (−1)i+ jM ij.
M (t) y1(t), . . . , ym(t)
xij i y j A = [aij]
dW
dt =
d
dt det M (t) =
mi=1
m j=1
∂ det M (t)
∂xij
dxijdt
=m
i,j=1(−1)i+ jM ij(t)
·
m
k=1 aik(t)xkj =m
i,k=1m
j=1(−1)i+ jaik(t)M ij(t)xkj(t)
=m
i,k=1
aik(t)m
j=1
(−1)i+ jM ij(t)xkj(t) = . . . .
j det M (t) i
k
. . . =
m
i,k=1 aik ·
W (t) k = i
0 k = i =m
i=1 aii(t)W (t) = trA(t) · W (t).
8/17/2019 Skrypt mat
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m
y(m) + am−1(t)y(m−1) + . . . + a0(t)y = 0,
dW
dt = −am−1(t)W (t).
Y (t)
Y (t) =
y
y
y(m−2)
y(m−1)
dY (t)dt =
y
y
y(m−1)
−am−1(t)y(m−1) − . . . − a0(t)y
.
A(t)
0 1 0 0 . . . 0
0 0 1 0 . . . 0
0 0 0 . . . 0 1
−a0 −a1 −a2 . . . −am−2 −am−1
−am−1
W (t0)
W (t) = W (t0) · expt
ˆ t0
trA du
W (t) = W (t0)exp
tˆ t0
−am−1(u) du.
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y(t) = A(t)y(t) + B(t)
M (t, u) M (u, u) = I
y(t) =
tˆ t0
M (t, u)B(u) du
dy
dt =
d
t
´ t0
M (t, u)B(u) du
dt = M (t, t)B(t) +
t
ˆ t0
d
dtM (t, u)B(u) du
= B(t) +
tˆ t0
A(t)M (t, u)B(u) du
= B(t) + A(t)
tˆ t0
M (t, u)B(u) du = B(t) + A(t)y
A
σ(A) Sp(A)
0 (f )
f (z ) =∞
k=0akz
k.
f (z ) B(0, (f )) (f ) = ∞
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A
λ1, . . . , λm q 1, . . . , q m
M k, 1 k m 0 q k − 1 f (z )
f (A) =mk=1
qk−1=0
M k, · df
dz (λk)
M k,
f (z ) = z r r = 0, . . . , m − 1
M k,l
Ar =mk=1
qk−1=1
M k, · r · . . . · (r − + 1)λr−k
cr
∀k,m−1r=0 r · . . . · (r − + 1)λ
r
−
k cr = 0
w(z ) =m−1r=0
crz r
∀k, w()(λk) = 0 λk w q k
w mk=1
q k = m w
m − 1
m
χA(A) = 0
χA(z ) A
χA(z ) = z m − wA(z ) deg wA < m
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r 0 z m+r z rwA(z )
z = A
z = λk 1
k
n
z = λk < q k
z rχA(z )
d
dz (z rχA(z )) =
s=0
s
(z r)(−s)(χA(z ))(s)
χ(s)A (λk) = 0 s λk q k − 1 s < q k
r 0 deg f < m + r r = 0 f = z m+r
f (A) = ArwA(A) deg wA m − 1
z rwA(z ) m + r
f (A) = ArwA(A) =mk=1
qk−1=0
M k,dl
dz (z rwA(z ))|z=λk =
mk=1
qk−1=0
M k,f ()(λk)
f
m+r
f m + r
f
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f (z ) f (z ) =∞n=0
cn(z − z 0)n
f (A) = limN →∞
N n=0
cn(A − z 0E )n
z 0 = 0
A λ1, . . . , λm q 1, . . . , q m
M k, 1 k m 0 q k − 1
f A
f (A) =mk=1
qk−1=0
M k, · df
dz (λk).
f (A) = limN →∞
N n=0
cnAn = lim
N →∞
N n=0
mk=1
qk−1=0
M k, · cn · n · . . . · (n − + 1)λn−k
=mk=1
qk−1=0
M k, · limN →∞
N n=0
cn · n · . . . · (n − + 1)λn−k =mk=1
qk−1=0
M k,f ()(λk)
λk A
E k := {v ∈ Rn : ∃q0 (A − λkE )qv = 0}
= 1
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Rm
Rm =
nk=1
E k
n
M k,
M k,
∀k Im M k, ⊂ E k k M k,0 E k
v =n
k=1
, vk ∈ E k ⇒ M k,0v = vk,
k = k 0 q k − 1 E k ⊂ ker M k,
dydt
= Ay A exp(At)
exp(0) = I
ddt
exp(At) = A exp(At)
f (z ) = exp(zt)
exp(At) =
nk=1
qk−1=0
M k,dezt
dz |z=λk =n
k=1
qk−1=0
M k, · t · etλk .
d
dt exp(At) =
nk=1
qk−1=0
M k,d
dt(t exp(tλk)) =
nk=1
qk−1=0
M k,(t + λkt
)exp(tλk).
g(z ) = z exp(tz )
A exp(At) = g(A) =n
k=1
qk−1=0
M k,dg(z )
dz |z=λk
dg
dz = zt exp(tz ) + t−1 exp(tz )
nk=1
qk−1=0
M k,dg(z )
dz |z=λk =
nk=1
qk−1=0
M k,(λkt + t−1)exp(tλk) =
d
dt exp(At).
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exp{At}
det
eA
= etr(A)
det
eA
= det (exp A · 1) = exp
det1 ·1ˆ
0
tr(A) du
= exp(tr(A)).
ψi = z i
i = 0, . . . ,n
j=1q j − 1
n
A p =n
k=1
qk−1=0
M k,dz p
dz (λk) =
nk=1
qk−1=0
M k, · p · ( p − 1) · . . . · ( p − l + 1)λ p−k .
c p k = 0 m−1 p=0
c pλ pk = 0
λk w(z ) =m−1 p=0
c pz p
0 =
m−1
p=0 c p p( p − 1) . . . ( p − + 1)λ
p−k =
m−1
p=0 c p p( p − 1) . . . ( p − + 1)w
()
(λk)
w q k
m − 1
eAt
λ̄(A) = max{Re λk : λk ∈ Sp(A)}.
̄(A) = max{ : 0 q k − 1, Re λk = λ̄(A), M k, = 0}.
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exp{At}
A = sup{Ax : x 1} A = [a jk ] max{|a jk| } A m max{|a jk|}
A λ̄(A) = λ̄ ̄(A) = ̄
0 0.
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a(t)y + b(t)y + c(t)y = 0
a,b,c
x(t) =∞n=0
xn(t − t0)n
x xn
xn = x(n)(t0)
n!
t0 a(t0) = 0
y + p(t)y + q (t)y = 0
p(t) = b(t)a(t)
q (t) = c(t)a(t)
t0
(t − t0)2y + (t − t0) p(t)y + q (t)y = 0.
t0
sin ty + cos ty + tg ty = 0.
t0 = 0
tsin t
t y + cos ty + tg ty = 0
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t2
sin t
t2y + t cos t t
sin ty + t tg t
t
sin ty = 0.
t0 = 0
t−t0 = z t0 = 0 y(t) y
y =∞k=0
cktk.
y
t
y(t) =∞k=1
kcktk−1 =
∞k=0
(k + 1)ck+1tk,
y(t) =∞k=2
k(k − 1)cktk−2 =∞k=0
(k + 1)(k + 2)ck+2tk.
∞
k=0 [ck+2(k + 2)(k + 1) + p(t)ck+1(k + 1) + q (t)ck] = 0. p(t) =
∞m=0
pmtm
q (t) =∞
m=0q mt
m
∞k=0
p(t)ck+1(k + 1)tk =
∞k=0
∞m=0
pmtmck+1(k + 1)t
k = . . .
= m + k m = − k
=∞
=0t
k=0ck+1 p−k(k + 1).
∞k=0
q (t)cktk =
∞=0
k=0
tckq −k.
∞=0
c+2( + 2)( + 1) +
k=0
ck+1(k + 1) p−k +
k=0
ckq −k
t ≡ 0
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c+2( + 2)( + 1) +
k=0
ck+1(k + 1) p−k +
k=0
ckq −k = 0.
ck
+ 2
c+2
+ 1
c+2 = −1
( + 1)( + 2)
k=0
ck+1(k + 1) p−k +
k=0
ckq −k
.
c0 = y(0) c1 = y(0)
y1 : c10 = 1, c
11 = 0,
y2 : c20 = 0, c
21 = 1,
y c0, c1
y
y + y = 0 p(t) = 0 q (t) = 1
c+2 = −1
( + 1)( + 2)c.
c0 = 1, c1 = 0 ⇒ c2k+1 = 0 c2 = −11·2 c4 = 11·2·3·4 cos t
c0 = 0, c1 = 0 ⇒ c2k = 0 c1 = 1 c3 = �