Линейное однородное уравнение: C постоянными

Ëèíåéíîå îäíîðîäíîå óðàâíåíèå:
L (y) =
n
X
(1)
ak y (k) = 0
k=0
L(y1 + y2 ) = L(y1 ) + L(y2 ),
Åñëè
(2)
L(αy) = αL(y).
L (y1 ) = 0 è L (y2 ) = 0,
(3)
L (αy1 + βy2 ) = αL (y1 ) + βL (y2 ) = 0.
(4)
y = C1 y1 + C2 y2 + . . . + Cn yn .
(5)
y = eλx
(6)
òî
Îáùåå ðåøåíèå:
Ñ ïîñòîÿííûìè êîýôôèöèåíòàìè
Èùåì ðåøåíèå â âèäå
L eλx =
n
X
ak eλx
(k)
=
k=0
n
X
ak λk eλx
k=0
Õàð. óðàâíåíèå:
Q (λ) =
n
X
1
= 0 · λx
e
(7)
(8)
ak λk = 0
k=0
Êîðíè êðàòíîñòè 1
(9)
y = C1 eλ1 x + . . . + Cn eλn x .
Êîðåíü λ = µ êðàòíîñòè m 6 n
Q (λ) =
n
X
m
ak λk = (λ − µ)
n−m
X
m
(10)
bk λk = (λ − µ) P (λ)
k=0
k=0
Ïóñòü m > j ∈ N. Íàéä¼ì
j
j
k=0
k=0
X
X
dj
m!
(j−k)
m (k)
(j−k)
m−k
Q (λ) =
Cjk (P (λ))
((λ − µ) ) =
Cjk (P (λ))
(λ − µ)
j
dλ
(m − k)!
(12)
k 6 j < m =⇒ m − k > 0
j
X
dj
m−k Q
(λ)
=
.
.
.
(λ
−
µ)
=0
dλj
λ=µ
λ=µ
(13)
k=0
Ïóñòü òåïåðü m > p ∈ N. Ïîêàæåì, ÷òî y = xp eµx ðåøåíèå óðàâíåíèÿ (1):
L (xp eµx ) =
n
X
k=0
ak (xp eµx )
(k)
=
n
X
k=0
1
ak
k
X
j=0
(j)
Ckj (xp )
(eµx )
(11)
(k−j)
=
=
n X
k
X
ak Ckj (xp )
(j)
µk−j eµx =
k=0 j=0
n X
n
X
ak
j=0 k=j
n
(j) n
X
(xp ) X
k!
k!
(j)
(xp ) µk−j eµx = eµx
ak
µk−j =
j! (k − j)!
j!
(k
−
j)!
j=0
k=j
n
(j) n
X
(xp ) X
dj k
µx
=
µ
=
e
ak
j!
dµj
j!
λ=µ
j=0
j=0
k=j
k=j



p
j−1
n
n
(j) j
j
p (j)
X
X
X
X
(xp )
d
(x
)
d
k 
µx
µx
k


=e
=e
Q (λ) ak λ +
ak λ
j
j
j!
dλ
j!
dλ
λ=µ
j=0
j=0
k=j
k=0
= eµx
n
(j) n
X
(xp ) X
ak
dj k λ dλj
(14)
λ=µ
j6p<m
L (xp eµx ) = eµx
p
(j)
X
(xp )
j!
j=0
y1 = eµx ,
y2 = xeµx ,
(15)
·0=0
(16)
ym = xm−1 eµx
...
÷òî è ò.ä.
Êîðåíü êîìïëåêñíûé: λ = α + iβ
λ = α − iβ òîæå êîðåíü
y1 = e(α+iβ)x = eαx (cos βx + i sin βx)
(17)
y2 = e(α−iβ)x = eαx (cos βx − i sin βx)
(18)
y = C1 e
= C1 e
αx
(α+iβ)x
+ C2 e
(cos βx + i sin βx) + C2 e
(α−iβ)x
αx
=
(19)
(cos βx − i sin βx) =
= (C1 + C2 ) eαx cos βx + (C1 − C2 ) eαx i sin βx
Ïóñòü C1 = a + ib, C2 = c + id. Âûäåëèì òàêèå C1 è C2 , ïðè êîòîðûõ ðåøåíèå y áóäåò äåéñòâèòåëüíûì.
Ïðè x = 0 :
y (0) = (C1 + C2 )
Ïðè x =
Im y (0) = Im (a + ib + c + id) = b + d = 0
(20)
d = −b
(21)
π
2β :
y
π
2β
πα
Im y
×àñòíûå ðåøåíèÿ:
πα
πα
= (C1 − C2 ) e 2β i = (a + ib − c − id) e 2β i = (ia − b − ic + d) e 2β
π
2β
πα
πα
= Im (ia − b − ic + d) e 2β = (a − c) e 2β = 0
(22)
(23)
c=a
(24)
C2 = a − ib = C1
(25)
y = (C1 + C2 ) eαx cos βx + (C1 − C2 ) eαx i sin βx = 2aeαx cos βx − 2beαx sin βx
(26)
y1 = eαx cos βx
y2 = eαx sin βx
2
(27)