В.П. Моденов - Дифференциальные уравнения (курс лекций), страница 5

Ìåòîäîì íåîïðåäåëåííûõ êîýèöèåíòîâ ðåøàåòñÿ íåîäíîðîäíîåóðàâíåíèå Ýéëåðà ñî ñïåöèàëüíîé ïðàâîé ÷àñòüþ f (x) = S(ln x)xλ (ïåðåõîäÿùåéïðè çàìåíå x = et â óíêöèþ f (t) = S(t)eλt ).Ïðèìåðû.1). y ′′ + 4y = e3xM (λ) = λ2 + 4 = 0, λ1,2 = ±2i 6= λ = 3, ỹ = Ae3x2). y ′′ + 4y = (x + 2)e3xM (λ) = λ2 + 4 = 0, λ1,2 = ±2i 6= λ = 3, ỹ = (Ax + B)e3x3). y ′′ − 4y = (x + 2)e2xM (λ) = λ2 − 4 = 0, λ1,2 = ±2, m1 = 1, λ = λ1 , ỹ = x(Ax + B)e2x4). y ′′′ + 3y ′′ + 3y ′ + y = (x + 2)e−x , M (λ) = (λ + 1)3 = 0, λ1 = −1, m1 = 3, λ = λ1 = −1ỹ = x3 (Ax + B)e−x5). y ′′ − 4y = cos 2x = Ree±2ix , M (λ) = λ2 − 4 = 0, λ1,2 = ±2, λ = ±2i, λ 6= λkỹ = A cos 2x + B sin 2x6).

y ′′ + 4y = cos 2x = Ree±2ix , M (λ) = λ2 + 4 = 0, λ1,2 = ±2i = λ, m1,2 = 1ỹ = x(A cos 2x + B sin 2x)33ëàâà 3Ñèñòåìû ëèíåéíûõ óðàâíåíèé.3.1Ñâîéñòâà ëèíåéíîé ñèñòåìû.Îïðåäåëåíèå 3.1. Ñèñòåìà ëèíåéíûõ óðàâíåíèé ⇐⇒Defd~y= A(x)~y (x) + f~(x),dx(3.1)ãäåf1 (x)y1 (x)°° .  , f~(x) =  . A(x) = °aij (x)° ,(i, j = 1, n), ~y (x) =  .  . fn (x)yn (x)aij (x), fi (x) ∈ C[a, b].(A)Ïîñòàâèâ íà÷àëüíîå óñëîâèå~y (x0 ) = ~y0 ,(3.2)ïîëó÷èì çàäà÷ó Êîøè (3.1), (3.2).Óòâåðæäåíèå.

Ïðè óñëîâèè (A), òî åñòü äëÿ ëèíåéíîé ñèñòåìû (3.1)ñ íåïðåðûâíûìè êîýèöèåíòàìè è ïðàâîé ÷àñòüþ ñïðàâåäëèâà òåîðåìàñóùåñòâîâàíèÿ è åäèíñòâåííîñòè ðåøåíèÿ çàäà÷è (3.1), (3.2) íà ïðîìåæóòêå[a, b] èõ íåïðåðûâíîñòè.Îïðåäåëåíèå 3.2. Åñëè f~(x) ≡ 0, òî ñèñòåìà ëèíåéíûõ óðàâíåíèéd~y= A(x)~y (x)dxíàçûâàåòñÿ îäíîðîäíîé, åñëè f~(x)íåîäíîðîäíîé.6=34(3.1′ )0, òî ñèñòåìà (3.1) íàçûâàåòñÿÒåîðåìà 3.1 (Ïðèíöèï ñóïåðïîçèöèè).

Ïóñòü:dy~1) dxj = A(x)~yj (x) + f~j (x), j = 1, N ;NPd~y2) dx= A(x)~y (x) + f~(x), f~(x) =Cj f~j (x) (Cj - ïîñòîÿííûå)j=1Òîãäà:NPCj ~yj (x) (Cj - òå æå ïîñòîÿííûå)~y =j=1ÄîêàçàòåëüñòâîNNNXXd~y X dy~jCj f~j (x) =CjCj ~yj (x) +== A(x)dxdxj=1j=1j=1= A(x)~y (x) + f~(x),÷.ò.ä.Ñëåäñòâèå 3.1. Ëèíåéíàÿ êîìáèíàöèÿ ðåøåíèé îäíîðîäíîé ñèñòåìû åñòüòàêæå ðåøåíèå ýòîé ñèñòåìû.Äîêàçàòåëüñòâî Ïîëîæèòü â Ò.3.1: f~j (x) ≡ 0, ∀j = 1, N .3.2Ëèíåéíàÿ îäíîðîäíàÿ ñèñòåìà.Îïðåäåëåíèå 3.3. {~yj (x), j = 1, n} - ëèíåéíî çàâèñèìû ∀x ∈ [a, b] ⇐⇒ ∃Cj Defïîñòîÿííûå,nPj=1|Cj | =6 0 (íå âñå ðàâíûå íóëþ):C1 ~y1 (x) + ... + Cj ~yj (x) + ... + Cn ~yn (x)≡∀x∈[a,b](*)~0,Îïðåäåëåíèå 3.4. {~yj (x), j = 1, n} - ëèíåéíî íåçàâèñèìû ∀x ∈ [a, b] ⇐⇒Def(∗) =⇒ C1 = C2 = ...

= Cj = ... = Cn = 0, ∀x ∈ [a, b].Îïðåäåëåíèå 3.5. Îïðåäåëèòåëü Âðîíñêîãî ⇐⇒Def¯¯ y11¯¯ ...¯detW (x) = detW [~y1 , ..., ~yj , ..., ~yn ] = ¯¯ ...¯ ...¯¯yn1... y1j... ...... ...... ...... ynj¯... y1n ¯¯... ... ¯¯... ... ¯¯... ... ¯¯... ynn ¯Òåîðåìà 3.2 (Íåîáõîäèìîå óñëîâèå ëèíåéíîé çàâèñèìîñòè). Åñëè{~yj (x), j = 1, n} - ëèíåéíî çàâèñèìû ∀x ∈ [a, b],òî ∀x ∈ [a, b] detW (x) = detW [~y1 , ..., ~yn ] ≡ 0.ÄîêàçàòåëüñòâîÒàê êàê ñòîëáöûïðîïîðöèîíàëüíû, òî detW (x) ≡ 0, ∀x ∈ [a, b].35~yjîïðåäåëèòåëÿÂðîíñêîãîÑëåäñòâèå 3.2 (Äîñòàòî÷íîå óñëîâèå ëèíåéíîé íåçàâèñèìîñòè). ∃x0 ∈[a, b], detW (x0 ) 6= 0 =⇒ {~yj , j = 1, n} - ëèíåéíî íåçàâèñèìû ∀x ∈ [a, b].Òåîðåìà3.3.ÎïðåäåëèòåëüÂðîíñêîãî ëèíåéíî íåçàâèñèìûõ ðåøåíèé îäíîðîäíîé ñèñòåìû äèåðåíöèàëüíûõóðàâíåíèé ñ íåïðåðûâíûìè êîýèöèåíòàìè îòëè÷åí îò íóëÿ.Äîêàçàòåëüñòâî (Îò ïðîòèâíîãî) Ïóñòü: ∃x0 ∈ [a, b] : detW (x0 ) = 0.Òîãäà îäíîðîäíîé ÑËÀÓ ñ îïðåäåëèòåëåì Âðîíñêîãî, ðàâíûì íóëþ,(**)C1 ~y1 (x0 ) + · · · + Cn ~yn (x0 ) = ~0C1 , C2 , ..., Cn :èìååòíåòðèâèàëüíîåðåøåíèåàññìîòðèì âåêòîð-óíêöèþ C1 ~y1 (x) + C2 ~y2 (x) + · · · + Cn ~yn (x) = ~y (x), êîòîðàÿâ ñèëó ñëåäñòâèÿ 3.1 Ò.3.1 è (??) ÿâëÿåòñÿ òðèâèàëüíûì ðåøåíèåì çàäà÷è Êîøè d~y = A(x)~y (x)dx⇔ ~y (x) = ~0 ~y (x ) = ~0(??)0⇐⇒ {~yj (x), j = 1, n} - ëèíåéíî çàâèñèìû, ÷òî ïðîòèâîðå÷èò óñëîâèþ òåîðåìû.DefÑëåäîâàòåëüíî, detW (x) 6= 0, ∀x ∈ [a, b], ÷.ò.ä.Îïðåäåëåíèå 3.6.

Ëèíåéíî íåçàâèñèìûå ðåøåíèÿ {~yj (x), j = 1, n} îäíîðîäíîé′ñèñòåìû (3.1 ) ⇐⇒ óíäàìåíòàëüíàÿ ñèñòåìà ðåøåíèé (ÔÑ) (3.1′ ), àDefñîîòâåòñòâóþùàÿ èì ìàòðèöà W (x) = W [~y1 , ..., ~yn ] ⇐⇒ óíäàìåíòàëüíàÿDefìàòðèöà (ÔÌ) îäíîðîäíîé ñèñòåìû.Ñëåäñòâèå 3.3. Îïðåäåëèòåëü Âðîíñêîãî èç ÔÑ îòëè÷åí îò íóëÿ.Îïðåäåëåíèå 3.6′ (Ýêâèâàëåíòíîå îïðåäåëåíèå ÔÌ).(10W (x) - ÔÌ ⇐⇒Def20dW (x)dx= A(x)W (x),detW (x) 6= 0, ∀x ∈ [a, b].Òåîðåìà 3.4 (Î ñóùåñòâîâàíèè ÔÑ). Âñÿêàÿ ëèíåéíàÿ îäíîðîäíàÿ ñèñòåìàóðàâíåíèé (3.1′ ) íåïðåðûâíûìè êîýèöèåíòàìè èìååò ÔÑ.Äîêàçàòåëüñòâî Çàäàäèì ïðîèçâîëüíûé ÷èñëîâîé îïðåäåëèòåëü, îòëè÷íûéîò íóëÿ¯¯¯ b11 · · · b1j · · · b1n ¯¯¯∆0 = ¯¯.

. . . . . . . . . . . . . . . . . .¯¯ 6= 0¯bn1 · · · bnj · · · bnn ¯Ïîñòðîèì n ðåøåíèé çàäà÷ Êîøèd~y j = A(x)~yj (x),{~yj (x), j = 1, n} : dxy~j (x0 ) = b~j36Ïîêàæåì, ÷òî ïîëó÷åííàÿ ñèñòåìà åñòü ÔÑ. Äëÿ ýòîãî äîñòàòî÷íî äîêàçàòü, ÷òî{~yj (x), j = 1, n} - ëèíåéíî íåçàâèñèìû.=⇒{~yj (x), j = 1, n}  ñàìîì äåëå, detW [y~1 (x0 ), ..., y~n (x0 )] = ∆0 6= 0Ñë.3.2 Ò.3.2ëèíåéíî íåçàâèñèìû ∀x ∈ [a, b]. Çíà÷èò, ïî îïðåäåëåíèþ 3.6 ñëåäóåò {~yj (x), j =1, n} - ÔÑ.Çàìå÷àíèå. Ò.ê. ∆0 6= 0 ñêîëüêî óãîäíî, òî ÔÑ áåñêîíå÷íî ìíîãî.Òåîðåìà 3.5 (Î ïîñòðîåíèè îáùåãî ðåøåíèÿ).

Ëþáîå ðåøåíèå îäíîðîäíîéñèñòåìû ñ íåïðåðûâíûìè êîýèöèåíòàìè ïðåäñòàâèìî â âèäå ëèíåéíîé~~~~ , ãäå⇐⇒ Z(x)= C1 y~1 + ... + Cny~n =W (x)Cêîìáèíàöèè ÔÑ: ddxZ = A(x)Z(x)C1 . ~ - íåêîòîðûé ïîñòîÿííûé âåêòîð-ñòîëáåö, C~ =  .W (x) - ÔÌ, C . Cn~- êàêîå-ëèáî ðåøåíèå çàäà÷è Êîøè:Äîêàçàòåëüñòâî Ïóñòü Z(x)(~dZdx~= A(x)Z(x),~ 0 ) = Z~0Z(x~ - ïîñòîÿííûé âåêòîð, òàêîé, ÷òî W (x)C~ = Z(x)∀x~Ïîêàæåì,÷òîª∃C∈ [a, b] (èëè©~∃ Cj , j = 1, n : C1 ~y1 + ...Cn ~yn ñîâïàäàåò ñ Z(x) ∀x ∈ [a, b]).~ óäîâëåòâîðÿåò òîé æå ñàìîé çàäà÷åÄëÿ ýòîãî äîñòàòî÷íî ïîêàçàòü, ÷òî W (x)C~Êîøè, ÷òî è Z(x)³´~³´d W (x)C~= A(x) W (x)Cdx~ = Z~0 ,W (x0 )C~ .à, çíà÷èò, â ñèëó òåîðåìû ñóùåñòâîâàíèÿ è åäèíñòâåííîñòè ñîâïàäàåò ñ Z(x)=⇒ ñàìîì äåëå, âî - ïåðâûõ,ëèíåéíàÿ êîìáèíàöèÿ ÔÑ óäîâëåòâîðÿåòÑë.3.1 Ò.3.1îäíîðîäíîé ñèñòåìå (3.1′ ).

Âî - âòîðûõ, íà÷àëüíîå óñëîâèå ýòîé çàäà÷è Êîøè~ . Ïðè x = x0 â ñèëó Ñë. 3.3 detW (x0 ) 6=ïðåäñòàâëÿåò ñîáîé ÑËÀÓ îòíîñèòåëüíî C~ = W −1 (x0 )Z~0 - ðåøåíèå ÑËÀÓ.0 =⇒ ∃W −1 (x0 ) è ∀~(Z0 )∃C~Ñëåäñòâèå 3.4. Âåêòîð-óíêöèÿ Z(x)~ , ãäå W (x) - ÔÌ, C~= W (x)C- ïðîèçâîëüíûé ïîñòîÿííûé âåêòîð, ÿâëÿåòñÿ îáùèì ðåøåíèåì (Î)îäíîðîäíîé ñèñòåìû~ dZ(x) = A(x)Z(x)~~~0dx⇔ Z(x)= W (x)W −1 (x0 )Z(***) ~Z(x0 ) = Z~0Îïðåäåëåíèå 3.7. Ìàòðèöà K(x, x0 ) = W (x)W −1 (x0 ) íàçûâàåòñÿ ìàòðèöåéÊîøè, ("èìïóëüñíîé"ìàòðèöåé èëè ìàòðèöàíòîì).37Òàê êàê K(x0 , x0 ) = W (x0 )W −1 (x0 ) = E , òî îíà îäíîçíà÷íî îïðåäåëÿåòñÿ êàêðåøåíèå çàäà÷è ÊîøèdK(x, x0 )= A(x)K(x, x0 ),dxK(x0 , x0 ) = E.Ñëåäîâàòåëüíî, â ñèëó (***)~ dZ = A(x)Z(x)~dx~Z(x0 ) = Z~03.3~~0.⇐⇒ Z(x)= K(x, x0 )ZÍåîäíîðîäíàÿ ëèíåéíàÿ ñèñòåìà.Òåîðåìà 3.6.

Ëþáîå ðåøåíèå ~y (x) íåîäíîðîäíîé ñèñòåìûd~y= A(x)~y (x) + f~(x)dx(3.3)ñ êîýèöèåíòàìè è ïðàâîé ÷àñòüþ ìîæíî ïðåäñòàâèòü â âèäå ñóììû îáùåãîðåøåíèÿ ñîîòâåòñòâóþùåé îäíîðîäíîé ñèñòåìû (ïðè f~(x) = 0) è íåêîòîðîãîðåøåíèÿ ýòîé íåîäíîðîäíîé ñèñòåìû~y (x) = W (x)C~0 + ~ỹ(x),(3.4)ãäå W (x) - ÔÌ, C~0 ïðîèçâîëüíûé ïîñòîÿííûé âåêòîð-ñòîëáåö.Äîêàçàòåëüñòâî Ò.ê. ~y(x) − ~ỹ(x) óäîâëåòâîðÿåò îäíîðîäíîé ñèñòåìå (3.1′ ),òî ïî Ò.3.5 ~y (x) − ~ỹ = W (x)C~0 .Òåîðåìà 3.7.d~y= A(x)~y (x) + f~(x),dx~y (x0 ) = y~0⇐⇒ ~y (x) = K(x, x0 )~y0 +ZxK(x, ξ)f~(ξ)dξx0Äîêàçàòåëüñòâî Èùåì ðåøåíèå íåîäíîðîäíîé ñèñòåìû óðàâíåíèé (3.3)ìåòîäîì âàðèàöèè ïîñòîÿííûõ (ÌÂÏ) â âèäå:~~y (x) = W (x)C(x),ãäå W (x) - ÔÌ⇒Def 3.6′ dW (x) − A(x)W (x) = 0,dx ∃W −1 (x).(3.5)Ïîäñòàâëÿÿ (3.5) â óðàâíåíèå (3.3), â ñèëó ïåðâîãî èç ýòèõ ñîîòíîøåíèé èìååìW (x)~dC(x)= f~(x),dx38÷òî â ñèëó âòîðîãî ñîîòíîøåíèÿ ðàâíîñèëüíî~dC(x)= W −1 (x)f~(x).dxÎòñþäà, èíòåãðèðóÿ, íàéäåì~C(x)= C~0 +ZxW −1 (ξ)f~(ξ)dξ.x0Ïîäñòàâëÿÿ íàéäåííîå çíà÷åíèå â (3.5), ïîëó÷èì~0 +~y (x) = W (x)CZxW (x)W −1 (ξ)f~(ξ)dξ =(3.6)x0~0 += W (x)CZxK(x, ξ)f~(ξ)dξ.x0Ïîëàãàÿ â (3.6) x = x0 , èç íà÷àëüíîãî óñëîâèÿ èìååì:~y (x0 ) = W (x0 )C~0 = y~0 .Îòêóäà íàõîäèìC~0 = W −1 (x0 )y~0W (x)C~0 = K(x, x0 )y~0 . ñèëó òåîðåìû åäèíñòâåííîñòè ðåøåíèÿ çàäà÷è Êîøè, äðóãèõ ðåøåíèé íåò.3.4Îäíîðîäíàÿ ñèñòåìà ëèíåéíûõ óðàâíåíèé ñïîñòîÿííûìè êîýèöèåíòàìè.àññìîòðèì îäíîðîäíóþ ñèñòåìó ëèíåéíûõ óðàâíåíèé ñ ïîñòîÿííûìèêîýèöèåíòàìèd~y= A~y (x),dx(3.7)ãäå A = kaij k , aij - ÷èñëà.(i,j=1,n)A.10 .Ñëó÷àé íåâûðîæäåííîãî ñïåêòðà îáñòâåííûõ çíà÷åíèé ìàòðèöûÒåîðåìà 3.8.

( Î ïîñòðîåíèè ÔÑ è îáùåãî ðåøåíèÿ îäíîðîäíîéñèñòåìû ñ ïîñòîÿííûìè êîýèöèåíòàìè.)Ïóñòü:{λj , j = 1, n} - íåâûðîæäåííûé ñïåêòð ñîáñòâåííûõ çíà÷åíèé A,{~αj , j = 1, n} - ñîîòâåòñòâóþùèå èì íåíóëåâûå ñîáñòâåííûå âåêòîðû A39© λxª~ j e j , j = 1, n - ÔÑ ñèñòåìû (3.7).Òîãäà: αÎáùåå ðåøåíèå (3.7) åñòü ëèíåéíàÿ êîìáèíàöèÿ ÔÑ:~y (x) =nXCj α~ j eλj x ,j=1ãäå Cj - ïðîèçâîëüíûå ïîñòîÿííûå.Äîêàçàòåëüñòâî Èùåì ðåøåíèå (3.7) â âèäå~y (x) = α~ eλxÏîäñòàâèâ â (3.7), ïîëó÷èì α~ λeλx = A~αeλx , îòêóäà èìååìÑËÀÓ:(A − λE)~α = ~0,(3.8)ãäå E - åäèíè÷íàÿ ìàòðèöà.Óñëîâèåì ñóùåñòâîâàíèÿ íåòðèâèàëüíîãî ðåøåíèÿ îäíîðîäíîé ÑËÀÓ (3.8)ÿâëÿåòñÿ ðàâåíñòâî íóëþ îïðåäåëèòåëÿ ìàòðèöû M (λ) = A − λE ýòîé ñèñòåìû:∃~α 6= ~0 ⇔ ∆(λ) = detM (λ) = det(A − λE) = 0 ⇔¯¯¯a11 − λ¯a...aa121,n−11,n¯¯¯ a21a22 − λ ...a2,n−1a2,n ¯¯¯⇔ ∆(λ) = ¯¯.