В.П. Моденов - Дифференциальные уравнения (курс лекций), страница 6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .¯¯ = 0¯ an−1,1 an−1,2 ... an−1,n−1 − λ an−1,n ¯¯¯¯ an,1an,2...an,n−1an,n − λ¯(3.9)Ìàòðèöà M (λ) = A − λE íàçûâàåòñÿ õàðàêòåðèñòè÷åñêîé , à óðàâíåíèå (3.9)∆(λ) = 0 - õàðàêòåðèñòè÷åñêèì óðàâíåíèåì . Êîìïîíåíòû ñîáñòâåííûõâåêòîðîâ α~j íàõîäÿòñÿ èç ðåøåíèÿ ÑËÀÓ (3.8) ñ òî÷íîñòüþ äî ïðîèçâîëüíîãîìíîæèòåëÿ.Çíàÿ êîðíè õàðàêòåðèñòè÷åñêîãî óðàâíåíèÿ ∆(λj ) = 0 è ðåøàÿ ÑËÀÓM (λj )~αj = ~0 (j = 1, n) (3.8) ìîæíî ïîñòðîèòü ñèñòåìó âåêòîð-ñòîëáöîâ:{α~j eλj x , j = 1, n}(*)Ñâîéñòâî 3.1.

Ñèñòåìà (*) - åñòü ñîâîêóïíîñòü ÷àñòíûõ ðåøåíèé (3.7), ÷òîñëåäóåò èç ïîñòðîåíèÿ.Ñâîéñòâî 3.2. Ñèñòåìà (*) - ëèíåéíî íåçàâèñèìûõ âåêòîð-ñòîëáöîâ, òàê êàê{α~j , j = 1, n} - ë.í. ⇔ {α~j eλj x , j = 1, n} - ë.í.Ñâîéñòâî 3.3. Ñèñòåìà (*) - îáðàçóåò ÔÑ (3.7), ÷òî ñëåäóåò èç ñâîéñòâ 3.1,3.2 è îïðåäåëåíèÿ ÔÑ.Ñâîéñòâî 3.4.

Ïî ë. Ò.2.5 îáùåå ðåøåíèå (3.7) åñòü ëèíåéíàÿ êîìáèíàöèÿ ÔÑ.Òåîðåìà äîêàçàíà.40Ïðèìåð 1.d~y= A~y , ãäåA =dxµ¶µ ¶0 1y, ~y = 11 0y2µ¶−λ 1Õàðàêòåðèñòè÷åñêàÿ ìàòðèöà M (λ) = A − λE =1 −λ¯¯¯−λ 1 ¯¯=0Õàðàêòåðèñòè÷åñêîå óðàâíåíèå ∆(λ) = 0 ⇐⇒ ¯¯1 −λ¯⇐⇒ λ2 − 1 = 0Ñîáñòâåííûå çíà÷åíèÿ A - ïðîñòûå λ1 = 1, λ2 = −1µ ¶1Ñ.ç. λ1 = 1 ñîîòâåòñòâóåò ñ.â.

α~1 =,1µ ¶1ñ.ç. λ2 = −1 ñîîòâåòñòâóåò ñ.â. α~2 =−1Ñëåäîâàòåëüíî, ÔÑ åñòüλ1 x{α~1 eλ2 x, α~2 e½µ ¶ µ ¶¾1 x1−x}, ò.å.e ,e.1−1Îáùåå ðåøåíèå:µ ¶µ ¶µ ¶11 xy1e−x .e + C2= C1−11y220 .Ñëó÷àé âûðîæäåííîãî ñïåêòðà îáñòâåííûõ çíà÷åíèé ìàòðèöû A.a) Àëãåáðàè÷åñêèé âåêòîðíûé ìåòîä.Óòâåðæäåíèå. Ïóñòü λk - ñ.ç. êðàòíîñòè mk ñîîòâåòñòâóåò mk ëèíåéíîíåçàâèñèìûõ âåêòîðîâ {~α1 , ..., α~ mk }, ñðåäè êîòîðûõ îäèí ñîáñòâåííûé âåêòîð, àîñòàëüíûå ïðèñîåäèíåííûå, à âñåì {λk , k = 1, l, m1 + m2 + ...

+ ml = n} îòâå÷àþòn ëèíåéíî íåçàâèñèìûõ âåêòîðîâ. Òîãäà ñèñòåìà âåêòîð-óíêöè骩 λx(**)α~ 1 e k , ..., α~ mk eλk x , k = 1, l, m1 + ... + ml = nóäîâëåòâîðÿåò ñâîéñòâàì 3.1 - 3.4, ÷òî è ñèñòåìà (*).Ïðèìåð 2.d~y= A~y , ãäåA =dxµ¶µ ¶3 0y, ~y = 1y20 3µ¶3−λ0Õàðàêòåðèñòè÷åñêàÿ ìàòðèöà M (λ) = A − λE =03−λ¯¯¯3 − λ0 ¯¯¯=0Õàðàêòåðèñòè÷åñêîå óðàâíåíèå ∆(λ) = 0 ⇐⇒ ¯03 − λ¯⇐⇒ (3 − λ)2 = 0Ñîáñòâåííîå çíà÷åíèå A - äâóõêðàòíîå λ1 = 3, m1 = 241Ñ.ç. λ = λ1 = 3 êðàòí.

m1 = 2 ñîîòâåòñòâóþò äâà ëèíåéíî íåçàâèñèìûõ âåêòîðñòîëáöà (ñîáñòâåííûé âåêòîð è ïðèñîåäèíåííûé âåêòîð):µ ¶µ ¶10α~1 =èα~2 =01ÔÑ åñòü:µ ¶ ¾½µ ¶1 3x 0 3xee ,10Îáùåå ðåøåíèå:µ ¶µ ¶µ ¶0 3x1 3xy1ee + C2= C110y2á) Ìåòîä íåîïðåäåëåííûõ êîýèöèåíòîâ (ÌÍÊ).Äëÿ ñ.ç. λk , (k = 1, l) êðàòíîñòè mk èùåì ëèíåéíî íåçàâèñèìûå ðåøåíèÿ â âèäå:a1 + a2 x + · · · + amk xmk −1y~k (x) = . .

. . . . . . . . . . . . . . . . . . . . . . . eλk xb1 + b2 x + · · · + bmk xmk −1Ïîäñòàâëÿÿ y~k (x) â ñèñòåìó (3.7), ñîêðàùàÿ íà eλk x 6= 0 è ïðèðàâíèâàÿêîýèöèåíòû ïðè îäèíàêîâûõ ñòåïåíÿõ x, ïîëó÷èì ÑËÀÓ èç mk îäíîðîäíûõóðàâíåíèé ñ mk × n íåèçâåñòíûìè.Èç îáùåãî ÷èñëà mk × n íåîïðåäåëåííûõ êîýèöèåíòîâ íåçàâèñèìûõêîýèöèåíòîâ áóäåò mk (êàêîâà êðàòíîñòü êîðíÿ).

Îñòàëüíûå êîýèöèåíòûâûðàæàþòñÿ ÷åðåç íèõ.Îáîçíà÷àÿ íåçàâèñèìûå ñâîáîäíûå êîýèöèåíòû ÷åðåç C1 , ..., Cmk èãðóïïèðóÿ ïðè íèõ ñëàãàåìûå, ïîëó÷èì:³´y~k = C1 P~1 (x) + . . . + Cmk P~mk (x) eλk x ,(k = 1, l; m1 + ... + ml = n),(***)ãäå P~j (x), (j = 1, mk ) - âåêòîð-ñòîëáöû, ó êîòîðûõ êàæäàÿ (êîîðäèíàòà)êîìïîíåíòà - êîíêðåòíûé ìíîãî÷ëåí ñòåïåíè 6 mk − 1Óòâåðæäåíèå. Ñèñòåìà (***) îáðàçóåò ÔÑ (3.7).

(Äîêàçàòåëüñòâî âó÷åáíèêå).Çàìå÷àíèå. Åñëè ñ.ç. λk êðàòíîñòè mk ñîîòâåòñòâóåò r ëèíåéíî íåçàâèñèìûõñ.â. A, òî ñòåïåíü ìíîãî÷ëåíîâ P~j (x) â (***) ìîæåò áûòü ïîíèæåíà äî mk − r.Ïðèìåð 3.µ¶d~y3 0= A~y , ãäåA =, λ1 = 3 - ñ.ç. êðàòíîñòè m1 = 2.0 3dxÈùåì ðåøåíèå â âèäå:~y (x) =µ¶a1 + a2 x 3xe .b1 + b2 x42Ïîäñòàâèâ â ÎÑ, ïîëó÷èì ÑËÀÓ èç 2x óðàâíåíèé ñ m1 × n = 4 íåèçâåñòíûìè:(a2 e3x + 3(a1 + a2 x)e3x = 3(a1 + a2 x)e3x ,b2 e3x + 3(b1 + b2 x)e3x = 3(b1 + b2 x)e3x(RangA = 2)Íåçàâèñèìûõ êîýèöèåíòîâ 2. Âûðàçèâ íåîïðåäåëåííûå êîýèöèåíòû ÷åðåçñâîáîäíûå êîíñòàíòû C1 , C2 :a1 = C1 , a2 = 0,b1 = C2 , b2 = 0.ðóïïèðóÿ ñëàãàåìûå ïðè C1 è C2 , ïîëó÷èì:µ ¶µ ¶1 3x0 3x~y = C1e + C2e .01|{z}|{z}P~13.5P~2Íåîäíîðîäíàÿ ñèñòåìà ëèíåéíûõ óðàâíåíèé ñïîñòîÿííûìè êîýèöèåíòàìè.àññìîòðèì ñèñòåìóãäåd~y= A~y (x) + f~(x),dx° °A = °aij ° , (i, j = 1, n).Ñïîñîáû íàõîæäåíèÿ ÷àñòíîãî ðåøåíèÿ íåîäíîðîäíîé ñèñòåìû:10 . Ìåòîäîì âàðèàöèè ïîñòîÿííûõ èëè ñ ïîìîùüþ ìàòðèöû Êîøè.20 .

Äëÿ ïðàâûõ ÷àñòåé ñïåöèàëüíîãî âèäà ("êâàçèïîëèíîìà") ïîäáîð ðåøåíèéìåòîäîì íåîïðåäåëåííûõ êîýèöèåíòîâ.Ñëåäñòâèå 3.5 (òåîðåìû 3.7).d~yZx= A~y (x) + f~(x),dx⇐⇒ ~y (x) = K(x − ξ)f~(ξ)dξ.~y (x0 ) = ~0x0Äîêàçàòåëüñòâî Ïîñòðîèì ìàòðèöó Êîøè äëÿ ñèñòåìû ñ ïîñòîÿííûìèêîýèöèåíòàìèdK(x, ξ)= AK(x, ξ),dxK(x, ξ) :K(ξ, ξ) = E.43Çàìåíà t = x − ξ ïðèâîäèò ê çàäà÷å Êîøè dK(t) = AK(t),´³dt=⇒ K = K(t) = Z~1 (t)...Z~n (t) , K(0) = EdZ~j= AZ~j (t),dt 0 ~ãäå Zj :: ~j (0) = 1 ← j ÿ ñòðîêà,Z  :0ñëåäîâàòåëüíî,Ïðèìåð 4.d~y= A~y + f~(x)dx~y (x0 ) = ~0, ãäåK = K(x − ξ), ÷.ò.ä.A=µ¶0 1,1 0Ïîñòðîèì ìàòðèöó Êîøè. Äëÿ ýòîãî ðåøèì äâå âåêòîðíûå çàäà÷è Êîøè.Íàéäåì âåêòîð-ñòîëáöû ìàòðèöû Êîøè ~ ~dZ2dZ1~1 (t)~=AZ dt = AZÃ2 (t) dtà !!Z~2 (t) : ~Z~1 (t) : ~10Z2 (0) =Z1 (0) =01(3.10)Âîñïîëüçîâàâøèñü ðåøåíèåì ïðèìåðà 1 è óäîâëåòâîðÿÿ íà÷àëüíûì óñëîâèÿì,ïîëó÷èì:µ ¶µ ¶µ¶1 t1C1 et + C2 e−t−t~Z1 (t) = C1e + C2e =,1−1C1 et − C2 e−tZ~1 (0) =µC1 + C2C1 − C2¶(µ ¶C1 + C2 = 11=⇐⇒0C1 − C2 = 0Ñëåäîâàòåëüíî,Z~1 (t) =µ1Z~2 (t) =µ1et + 21 e−t21 te − 21 e−t2¶=µ¶chtsht=µ¶shtchtÀíàëîãè÷íî,et21 te2− 21 e−t+ 21 e−t44¶1=⇒ C1 = C2 = .2Èòàê, ìàòðèöà Êîøè èìååò âèä:¶¶µµch(x − ξ) sh(x − ξ)cht sht⇐⇒ K(x − ξ)K(t) =sh(x − ξ) ch(x − ξ)sht chtÑëåäîâàòåëüíî, ðåøåíèå èñõîäíîé çàäà÷è Êîøè äëÿ íåîäíîðîäíîé ñèñòåìûóðàâíåíèé åñòü:~y (x) =ZxZx µK(x − ξ)f~(ξ)dξ ⇐⇒ ~y (x) =x0x03.6¶ch(x − ξ) sh(x − ξ) ~f (ξ)dξsh(x − ξ) ch(x − ξ)Î ðåøåíèè "âåêîâîãî óðàâíåíèÿ".àññìîòðèì õàððàêòåðèñòè÷åñêîå óðàâíåíèå (3.9)∆(λj ) = 0,ãäåλj- ïðîñòûå êîðíè.Ïðèìåíèìáèíàðíûéèòåðàöèîííûéäèåðåíöèàëüíî-ïàðàìåòðè÷åñêèìêîððåêòîðìåòîäîì-ïðîöåññ.(ÄÏ-ìåòîäîì)Áóäåìñâû÷èñëÿòüóòî÷íåíèåìλjìåòîäîìÍüþòîíà:λk+1j=λkj−∆(λkj )∆′ (λkj ).Âûáîð íà÷àëüíîãî ïðèáëèæåíèÿ îñóùåñòâëÿåì ÄÏ - ìåòîäîì.Ââåäåì ïàðàìåòðt ∈ [0, 1]:D(λj , t) = 0.àññìîòðèì óðàâíåíèåÑ÷èòàåìλj = λj (t)- óíêöèåé ïàðàìåòðàtíåÿâíîçàäàííîé, ïðè÷åì ïðèt = 0: D(λ0j , 0) = 0, λ0j - èçâåñòíû;t = 1: D(λj , 1) ≡ ∆(λj ) = 0.Ïî òåîðåìå î äèåðåíöèðîâàíèè íåÿâíî çàäàííîé óíêöèè íàõîäèìλj = λj (1)ïðèðåøåíèè çàäà÷è Êîøè:dλjD′= − ′t ,dtD λjλj |t=0 = λ0j .Íàéäåííîå çíà÷åíèå êîðíÿ ïðèíèìàåì çà íà÷àëüíîå ïðèáëèæåíèå, åñëè îíî âõîäèò âîáëàñòü ñõîäèìîñòè ìåòîäà Íüþòîíà.45ëàâà 4Êðàåâàÿ çàäà÷à äëÿ íåîäíîðîäíîãîäèåðåíöèàëüíîãî óðàâíåíèÿâòîðîãî ïîðÿäêà.4.1Ôîðìóëà Ëèóâèëëÿ - Îñòðîãðàäñêîãî.Îïðåäåëåíèå 4.1.

Îïåðàòîð Øòóðìà - Ëèóâèëëÿ ⇐⇒Def·¸dydp(x)− q(x)y(x),L[y] =dxdxãäå 0 < p(x) ∈ C1 , q(x) ∈ C.Ëåììà 4.1 (Ôîðìóëà Ëèóâèëëÿ - Îñòðîãðàäñêîãî). Ïóñòü: y1 (x) è y2 (x) -äâà ëèíåéíî íåçàâèñèìûõ ðåøåíèÿ îäíîðîäíîãî óðàâíåíèÿL[y] = 0误yW [y1 , y2 ] = ¯¯ dy11dxîïðåäåëèòåëü Âðîíñêîãî.Òîãäà:¯dy2dy1y2 ¯¯− y2dy2 ¯ = y1dxdx(6= 0)dxp(x)W [y1 , y2 ] = C,(*)ãäå C - êîíñòàíòà, íåðàâíàÿ íóëþ.Äîêàçàòåëüñòâî Ïîëó÷èì âñïîìîãàòåëüíîå òîæäåñòâî äëÿ ðåøåíèé y1 è y2 ,óäîâëåòâîðÿþùèõ îäíîðîäíûì óðàâíåíèÿì:µ¶ddy1p− qy1 = 0,dxdxµ¶dy2dp− qy2 = 0.dxdx46Óìíîæèâ ïåðâîå óðàâíåíèå íà (−y2 ), âòîðîå - íà (y1 ) è ñëîæèâ, áóäåì èìåòü:µ¶µ¶ddy2ddy1y1(**)p− y2p=0dxdxdxdxÄëÿ äîêàçàòåëüñòâà (*) äîñòàòî÷íî ïîêàçàòü, ÷òî ïðîèçâîäíàÿ ëåâîé ÷àñòè (*)ðàâíà íóëþ, ò.å.· µ¶¸id hddy1dy2− y2pW [y1 , y2 ] =p y1= 0 (òîæäåñòâî Ëàãðàíæà)dxdxdxdx· µ¶¸· µ¶¸ddy2dy1dy1 p−y2 p=⇐⇒dxdxdxdxµ¶µ¶µ¶µ¶dy2dy2dy1dy2dy1ddy1dp−pp− y2p=+ y1= 0.dxdxdxdxdxdxdxdx|{z} |{z}=04.2=0(∗∗)åøåíèå êðàåâîé çàäà÷è ìåòîäîì óíêöèèðèíà.àññìîòðèì ïåðâóþ êðàåâóþ çàäà÷ó:L[y] = f (x),(a)y(0) = 0,y(l) = 0,(á)(4.1)ãäå 0 < p(x) ∈ C1 (0, l); q(x), f (x) ∈ C[0, l].åøåíèåì äàííîé êðàåâîé çàäà÷è áóäåì ñ÷èòàòü äâàæäû äèåðåíöèðóåìóþóíêöèþ, óäîâëåòâîðÿþùóþ óðàâíåíèþ è ãðàíè÷íûì óñëîâèÿì (à) è (á).Ïóñòü ñîîòâåòñòâóþùàÿ îäíîðîäíàÿ çàäà÷à èìååò òîëüêî íóëåâîåðåøåíèå:L[y] = 0,(4.1′ )y(0) = 0, (a) ⇐⇒ y(x) ≡ 0.y(l) = 0, (á)Òîãäà, â ñèëó ëèíåéíîñòè ðàññìàòðèâàåìàÿ êðàåâàÿ çàäà÷à (4.1) èìååòåäèíñòâåííîå ðåøåíèå.Ïîñòðîèì ýòî ðåøåíèå ìåòîäîì óíêöèè ðèíà.01 .

Ïóñòü èçâåñòíû äâà íåòðèâèàëüíûõ ðåøåíèÿ äâóõ çàäà÷ Êîøè äëÿ îäíîðîäíîãîóðàâíåíèÿ (4.1′ ) ñîîòâåòñòâåííî ñ íà÷àëüíûìè óñëîâèÿìè (à) è (á)((L[y1 ] = 0L[y2 ] = 0y1 (x) 6= 0 :y2 (x) 6= 0 :y1 (0) = 0 (a)y2 (l) = 0 (á)Òîãäà:L[y1 ] = 0,y1 (l) 6= 0, èíà÷å y1 (0) = 0,y1 (l) = 047=⇒ y1 ≡ 0;L[y2 ] = 0,y2 (0) 6= 0, èíà÷å y2 (0) = 0,y2 (l) = 0=⇒ y2 ≡ 0.20 . åøåíèÿ ðàññìàòðèâàåìûõ çàäà÷ Êîøè y1 (x) è y2 (x) - ëèíåéíî íåçàâèñèìû∀x ∈ [0, l].Äåéñòâèòåëüíî, åñëè áû y1 è y2 áûëè ëèíåéíî çàâèñèìû, íàïðèìåð,y1 = Cy2(C 6= 0),òî y1 (l) = Cy2 (l) = 0, ÷òî ïðîòèâîðå÷èò 10 , òàê êàê y1 (l) 6= 0.30 . Èùåì ðåøåíèå óðàâíåíèÿ (4.1), èëè åìó ðàâíîñèëüíîãî óðàâíåíèÿy ′′ +p′ ′ qf (x)y − y== f1 (x),ppp(4.2)(ðàçðåøåí. îòí.

ñòàðøåé ïðîèçâîäíîé)ìåòîäîì âàðèàöèè ïîñòîÿííûõ â âèäåy(x) = C1 (x)y1 (x) + C2 (x)y2 (x).(4.3)Ïîëó÷èì ÑËÀÓ äëÿ C1′ (x) è C2′ (x):(C1′ (x)y1 (x) + C2′ (x)y2 (x) = 0,C1′ (x)y1′ (x) + C2′ (x)y2′ (x) = f1 (x).Îïðåäåëèòåëü ñèñòåìû åñòü îïðåäåëèòåëü Âðîíñêîãî: ∆ = W 6= 0, ò.ê.ïî äîêàçàííîìó â 20 Ôóíêöèè y1 , y2 - ëèíåéíî íåçàâèñèìû. Ñëåäîâàòåëüíî,ñóùåñòâóåò åäèíñòâåííîå ðåøåíèå ÑËÀÓ.