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)