В.П. Моденов - Дифференциальные уравнения (курс лекций), страница 4

+ Cn yn ñàìîìäåëå , ª©yk (x), k = 1, nðåøåíèå ÑËÀÓ:- ÔÑ⇒Ñë2.3 T 2.3W (x0 ) 6= 0 ⇒ ñóùåñòâóåò åäèíñòâåííîå~ 0 6= ~0.∃ (C1 , C2 , ..., Cn ) = CÔóíêöèÿnPCk yk (x) óäîâëåòâîðÿåò òîé æå çàäà÷å Êîøè (**), ÷òîk=1è óíêöèÿ Z = Z(x), ñëåäîâàòåëüíî, â ñèëó òåîðåìû ñóùåñòâîâàíèÿ èåäèíñòâåííîñòè ðåøåíèÿ ýòîé çàäà÷ènXCk yk (x) = Z,k=1ò.å. ëþáîå ðåøåíèå îäíîðîäíîãî óðàâíåíèÿ ìîæíî ïðåäñòàâèòü â âèäå ëèíåéíîéêîìáèíàöèè ÔÑ, ÷.ò.ä.26Ñëåäñòâèÿ Ò.2.5.1 . Ëèíåéíàÿ êîìáèíàöèÿ ÔÑ: y =0nPCk yk (x), ãäåk=1©yk (x), k = 1, nª- ÔÑ,ÿâëÿåòñÿ îáùèì ðåøåíèåì óðàâíåíèÿ Ly = 0.20 . Ìíîæåñòâî ðåøåíèé îäíîðîäíîãî ëèíåéíîãî óðàâíåíèÿ n ïîðÿäêà îáðàçóåòëèíåéíîå ïðîñòðàíñòâî .

( ñèëó Ñë.2.1 Ò.2.1 ëèíåéíàÿ êîìáèíàöèÿ ðåøåíèéñíîâà åñòü ðåøåíèå.)30 . Ëþáàÿ ÔÑ îáðàçóåò áàçèñ â ýòîì ïðîñòðàíñòâå. Ñëåäîâàòåëüíî, îíî n ìåðíî (â ñèëó îïðåäåëåíèÿ ÔÑ è Ò.2.5).2.3Íåîäíîðîäíîå ëèíåéíîå óðàâíåíèå.Òåîðåìà 2.6. Ëþáîå ðåøåíèå íåîäíîðîäíîãî óðàâíåíèÿLy = f (x)(2.5)Lỹ = f (x)(2.5')åñòü ñóììà åãî ÷àñòíîãî ðåøåíèÿè îáùåãî ðåøåíèÿ ñîîòâåòñòâóþùåãî îäíîðîäíîãî (f (x) ≡ 0), òî åñòüy = ỹ +nXCk yk (x)k=1ª©ãäå yk (x), k = 1, n - ÔÑ, Ck - íåêîòîðûå ïîñòîÿííûå.Äîêàçàòåëüñòâî Âû÷èòàÿ èç óðàâíåíèÿ (2.5) óðàâíåíèå (2.5') ïîëó÷èì L(y−ỹ) = 0. Îòêóäà â ñèëó Ò.2.5 ⇒ y − ỹ =nPCk yk (x),÷.ò.ä.k=1"Íà ñâåòå åñòü âåùè ïîâàæíåå ñàìûõ ïðåêðàñíûõ îòêðûòèé - ýòî çíàíèåìåòîäà, êîòîðûì îíî áûëî ñäåëàíî".

Ëåéáíèö.Äâà ìåòîäà ðåøåíèÿ íåîäíîðîäíîãî ëèíåéíîãî óðàâíåíèÿ.1 . Ìåòîä âàðèàöèè ïîñòîÿííûõ.20 . Ìåòîä óíêöèè Êîøè.10 . Ìåòîä âàðèàöèè ïîñòîÿííûõ (ÌÂÏ).Èùåì ðåøåíèå íåîäíîðîäíîãî óðàâíåíèÿ0Ly = f (x)27â âèäå ëèíåéíîé êîìáèíàöèè ÔÑ ñ íåêîòîðûìè óíêöèÿìè Ck (x):an (x)× y =an−1 (x)× y ′ =nPCk (x)yk (x)k=1nPk=1′′an−2 (x)× y =nPCk (x)yk′ (x) +k=1+(∗)Ck (x)yk′′ (x)nXCk′ (x)yk (x)|k=1+nX{z}{z}=0Ck′ (x)yk′ (x)|k=1=01× y(n)nP=k=1(n)Ck (x)yk (x)+Ly =|k=1{z=0(n−1)Ck′ (x)yk|k=1Ñóììèðóÿ, ïîëó÷èìnXnX|k=1{z(x)}=f (x)(n + 1)} ......... ...........................................................nnXP(n−1)(n−2)Ck (x)yk(x) +Ck′ (x)yk(x)a1 (x)× y (n−1) =k=1=0nz}|{ X(n−1)(x) = f (x),Ck′ (x)ykCk (x) Lyk +{z=0}|k=1{z=f (x)}ãäå äëÿ íàõîæäåíèÿ ïðîèçâîäíûõ èñêîìûõ óíêöèé èìååì ÑËÀÓ: nP ′C (x)yk (x) = 0, k=1 k.................................nP(n−1)Ck′ (x)yk(x) = f (x).k=1Îïðåäåëèòåëü ÑËÀÓ: ∆ = W (x) 6= 0 ⇒ ∃ åäèíñòâåííîå ðåøåíèå ÑËÀÓ {Ck′ (x)}:RxCk′ (x) = ϕk (x) - èçâåñòíûå óíêöèè ⇒ Ck (x) = ϕk (ξ)dξ + Ck .x0Ïîäñòàâëÿÿ â èñêîìûé âèä ðåøåíèÿ (*), ïîëó÷èìy=nX+Ck yk (x)}|k=1 {zîáùåå ðåøåíèåîäíîðîäíîãî óðàâíåíèÿ20 .

Ìåòîä óíêöèè Êîøè.28nXk=1yk (x)Zxxϕk (ξ)dξ{z0}|÷àñòíîå ðåøåíèå Ly = fñ íóëåâûìè íà÷. óñë..Îïðåäåëåíèå 2.9. Ôóíêöèÿ Êîøè ("èìïóëüñíàÿ"óíêöèÿ)çàäà÷è Êîøè- ðåøåíèåLx K(x, ξ) = 0LK(x,ξ)=0Ly(x)=0xK(ξ,ξ)=0y(ξ)=0 K(x, x) = 0′′Kx′ (x, x) = 0Kx (ξ, ξ) = 0y (ξ) = 0⇔K(x, ξ)= y(x) :⇔.......................................(x, ξ) ∈ [a, b] (n−1) (n−1) (n−1)y(ξ) = 1ξ - ïàðàìåòðKx(x, x) = 1Kx(ξ, ξ) = 1Ïðèìåð 1.½Ïðèìåð 2.y ′ (x) − y(x) = 0y(ξ) = 1 ′′ y (x) + y(x) = 0y(ξ) = 0 ′y (ξ) = 1Òåîðåìà 2.7. Ôóíêöèÿ y =Rx⇔ y = K(x, ξ) = ex−ξ⇔ y = K(x, ξ) = sin (x − ξ)K(x, ξ)f (ξ)dξ ÿâëÿåòñÿ ðåøåíèåì çàäà÷è Êîøè äëÿx0íåîäíîðîäíîãî óðàâíåíèÿ ñ íóëåâûìè íà÷àëüíûìè óñëîâèÿìè:½Ly = f (x),y(x0 ) = y ′ (x0 ) = ...

= y (n−1) (x0 ) = 0ãäå K(x, ξ) - óíêöèÿ Êîøè ñîîòâåòñòâóþùåãî óðàâíåíèÿ ïðè f (x) ≡ 0 x, x0 , ξ ∈[a, b].Äîêàçàòåëüñòâî (ïðîâåðêîé:)an (x)× y =Rxíà÷àëüíûå óñëîâèÿâûïîëíåíû:K(ξ)f (ξ)dξ=⇒ y(x0 ) = 0x0Rxan−1 (x)× y ′ = K(x, x)f (x) + Kx′ (ξ)f (ξ)dξ|{z} x0=0′′an−2 (x)× y =Kx′ (x, x)f (x) +{z|}=0RxKx′′ (ξ)f (ξ)dξ=01× y=Kx(n−1) (x, x)f (x) +|Ñóììèðóÿ, ïîëó÷èì{z}=f (x)Ly = f (x) +=⇒ y ′′ (x0 ) = 0x0................ ...........................................................Rx (n−1)a1 (x)× y (n−1) = Kx(n−2) (x, x)f (x) + Kx(ξ)f (ξ)dξ{z} x0|(n)=⇒ y ′ (x0 ) = 0Zxx0Rx=⇒ y (n−1) (x0 ) = 0(n)Kx (x, ξ)f (ξ)dξx0L K(x, ξ) f (ξ)dξ = f (x), ÷.ò.ä.| x {z }=0 ñèëó òåîðåìû åäèíñòâåííîñòè äðóãèõ ðåøåíèé íåò.29Ïðèìåð 3.½y ′ (x) − y(x) = f (x)⇔ y(x) =y(0) = 0Zxex−ξ f (ξ)dξ0Ïðèìåð 4. ′′Zx y (x) + y(x) = f (x)y(0) = 0⇔ y = sin (x − ξ)f (ξ)dξ ′y (0) = 00Çàìå÷àíèå.Ly(x) = δ(x − ξ0 )y(x0 ) = 0y ′ (x0 ) = 0.............y (n−1) (x0 ) = 0⇔y=ZxK(x, ξ)δ(ξ − ξ0 )dξx0=K(x, ξ0 ) ïî îïð.δ --öèèóíêöèÿ âëèÿíèÿ ìãíîâåííîãî åäèíè÷íîãî èñòî÷íèêà ("èìïóëüñíàÿ"óíêöèÿ),ñîñðåäîòî÷åííîãî â ò.

ξ0 , íà òî÷êó x.2.4Ëèíåéíûå óðàâíåíèÿêîýèöèåíòàìè.ñïîñòîÿííûìè2.4.1. Îäíîðîäíîå óðàâíåíèå.Îïðåäåëåíèå 2.10. Ëèíåéíîå îäíîðîäíîå óðàâíåíèå ñ ïîñòîÿííûìèêîýèöèåíòàìè ⇐⇒defLy = y (n) + a1 y (n−1) + ... + an y = 0, ãäå ai = const, i = 1, n.(2.6)Èùåì ÷àñòíûå ðåøåíèÿ óðàâíåíèÿ (2.6) â âèäå eλx (Ýéëåð). Ïîäñòàâëÿÿ â (2.6),ïîëó÷èì:¡ ¢ £¤L eλx = λn + a1 λn−1 + ... + an eλx = M (λ)eλx ⇔¡ ¢L eλx = M (λ)eλx(*)Îïðåäåëåíèå 2.11.

Ìíîãî÷ëåíM (λ) = λn + a1 λn−1 + ... + an30íàçûâàþò õàðàêòåðèñòè÷åñêèì ìíîãî÷ëåíîì óðàâíåíèÿ (2.6), à óðàâíåíèåM (λ) = λn + a1 λn−1 + ... + an = 0(2.7)íàçûâàþò õàðàêòåðèñòè÷åñêèì óðàâíåíèåì äëÿ (2.6).àññìîòðèì äâà ñëó÷àÿ.2.4.1.1. Õàðàêòåðèñòè÷åñêîå óðàâíåíèå (2.7) èìååò n ðàçëè÷íûõ êîðíåé λ1 ,λ2 ,... λn : M (λk ) = 0. Êîðíþ λk ñîîòâåòñòâóåò óíêöèÿ yk = eλk x , (k = 1, n)Ñâîéñòâà óíêöèé©y1 = eλ1 x , y2 = eλ2 x , ..., yn = eλn x1). Ñèñòåìà (2.8) åñòü ñèñòåìà ÷àñòíûõ ðåøåíèé (2.6).Äîêàçàòåëüñòâî  ñèëó òîæäåñòâà (*):ª(2.8)¡¢L eλk x = M (λk )eλk x = 0, ò.ê.

M (λk ) = 0, ÷.ò.ä.2). Ñèñòåìà (2.8) åñòü ñèñòåìà ëèíåéíî íåçàâèñèìûõ óíêöèé.Äîêàçàòåëüñòâî Îïðåäåëèòåëü Âðîíñêîãî:¯¯¯¯ y1y...y2n¯¯ ′′′ ¯¯ y1y2...yn ¯¯=W [y1 , y2 , ..., yn ] = ¯............ ¯¯¯ (n−1)(n−1) ¯(n−1)¯y... yny21¯¯ λx¯ e 1eλ2 x...eλn x ¯¯¯¯ λ eλ1 xλ2 eλ2 x ... λn eλn x ¯¯== ¯¯ 1............ ¯¯¯ n−1¯λ1 eλ1 x λ2n−1 eλ2 x ... λn−1 eλn x ¯¯¯ n¯ 11...1 ¯¯¯¯ λ1λ2 ... λn ¯¯ (λ1 +λ2 +...+λn )x= ¯¯e=... ... ... ¯¯¯ ...¯λ1n−1 λ2n−1 ... λn−1 ¯n|{z}îïðåäåëèòåëü Âàíäåðìîíäà= (λ1 − λ2 )(λ1 − λ3 )...(λ1 − λn )(λ2 − λ3 )...(λ2 − λn )......................(λn−1 − λn ) ×e(λ1 +λ2 +...+λn )x 6= 0,åñëè êîðíè óðàâíåíèÿ (2.7) ðàçëè÷íû.3).

Ñèñòåìà (2.8) îáðàçóåò ÔÑ óðàâíåíèÿ(2.6).©ªÄîêàçàòåëüñòâî Èç 1)., 2). ⇒ yk , k = 1, n - ÔÑ.Def4). Îáùåå ðåøåíèå (2.6): y(x) =nPCk eλk x .k=1Çàìå÷àíèå.  ñëó÷àå êîìïëåêñíûõ êîðíåé ïàðó êîìïëåêñíîçíà÷íûõ óíêöèée(α+iβ)x , e(α−iβ)x çàìåíÿþò âåùåñòâåííûìè óíêöèÿìè eαx cos βx, eαx sin βx èïîëó÷àþò äðóãóþ ÔÑ.2.4.1.2. Óðàâíåíèå M(λ) = 0 (2.7) èìååò êðàòíûå êîðíè :M (λ) = (λ − λ1 )m1 (λ − λ2 )m2 ... (λ − λk )mk ... (λ − λl )ml = 031ãäå mk - êðàòíîñòü λk , m1 + m2 + ... + mk + ... + ml = n. ñèëó îðìóëû Ëåéáíèöà èìååò ìåñòî òîæäåñòâî:¡¢ ©ªL xp eλx = xp M (λ) + pxp−1 M ′ (λ) + ... + M (p) (λ) eλxÊîðíþ λk êðàòíîñòè mk óðàâíåíèÿ (2.6) ñîîòâåòñòâóþò mk óíêöèé:ª© λx λxe k , xe k , ..., xmk −1 eλk x ; k = 1, l(**)(2.9)Ñâîéñòâà:1).

Ñèñòåìà óíêöèé (2.9) åñòü ñèñòåìà ÷àñòíûõ ðåøåíèé (2.6)Äîêàçàòåëüñòâî λk êîðåíü êðàòíîñòè mk ⇔ M (λk ) = 0, M ′ (λk ) =Def0, ¡..., M ¢ (λk ) = 0, M(λk ) 6= 0. Ñëåäîâàòåëüíî, â ñèëó òîæäåñòâà (**)p λk xL xe= 0, ∀p = 0, 1, ..., mk − 1; k = 1, l.2). Ñèñòåìà óíêöèé (2.9) ëèíåéíî íåçàâèñèìà (ñì. ó÷åáíèê ).3). Ñèñòåìà óíêöèé (2.9) îáðàçóåò ÔÑ (ïî îïðåäåëåíèþ ÔÑ).4). Îáùåå ðåøåíèå (2.6) åñòü ëèíåéíàÿ êîìáèíàöèÿ ÔÑ (Ò.2.5).Çàìå÷àíèå.

 ñëó÷àå êîìïëåêñíûõ êîðíåé ïàðó êîìïëåêñíîçíà÷íûõóíêöèé xp e(α+iβ)x , xp e(α−iβ)x çàìåíÿþò âåùåñòâåííûìè óíêöèÿìè xp eαx cos βx,xp eαx sin βx è ïîëó÷àþò äðóãóþ ÔÑ.(mk −1)(mk )2.4.2. Íåîäíîðîäíîå óðàâíåíèå. Ìåòîä íåîïðåäåëåííûõ êîýèöèåíòîâ.àññìîòðèì óðàâíåíèå ñ ïîñòîÿííûìè êîýèöèåíòàìèLy = y (n) + a1 y (n−1) + ... + an y = f (x),(2.10)ãäå f (x) = S(x)eλx , S(x) - ìíîãî÷ëåí ñòåïåíè s, λ - êîíñòàíòà.Óòâåðæäåíèå. Ïóñòü λ1 , λ2 , ..., λk , ..., λl - êîðíè õàðàêòåðèñòè÷åñêîãîóðàâíåíèÿ M (λ) = 0 êðàòíîñòåé m1 , m2 , ..., mk , ..., ml , ãäå m1 + m2 + ... + mk +... + ml = n.Òîãäà:1).

Åñëè λ 6= λk (k = 1, ..., l) (íåðåçîíàíñíûé ñëó÷àé), òî ÷àñòíîå ðåøåíèåóðàâíåíèÿ (2.10) èùåì â âèäåỹ(x) = Ps (x)eλx ,ãäå Ps (x) - ìíîãî÷ëåí ñòåïåíè s, ñ íåîïðåäåëåííûìè êîýèöèåíòàìè.2). Åñëè λ = λk (êðàòí. mk ) (ðåçîíàíñíûé ñëó÷àé), òî ÷àñòíîå ðåøåíèå óðàâíåíèÿ(2.10) èùåì â âèäåỹ(x) = xmk Qs (x)eλx ,ãäå Qs (x) - ìíîãî÷ëåí ñòåïåíè s ñ íåîïðåäåëåííûìè êîýèöèåíòàìè.Ïîäñòàâëÿÿ èñêîìûé âèä ðåøåíèé â (2.10) è ïðèðàâíèâàÿ êîýèöèåíòû ïðèîäèíàêîâûõ ñòåïåíÿõ x, íàõîäèì íåîïðåäåëåííûå êîýèöèåíòû ìíîãî÷ëåíîâPs (x) è Qs (x) (ìåòîä íåîïðåäåëåííûõ êîýèöèåíòîâ ).Çàìå÷àíèå 1. Ê óðàâíåíèþ ñ ïîñòîÿííûìè êîýèöèåíòàìè ñâîäèòñÿ îäíîðîäíîåóðàâíåíèå Ýéëåðàxn y (n) + a1 xn−1 y (n−1) + ... + an y = 0,32åñëè ïîëîæèòü x = et .Çàìå÷àíèå 2.