В.П. Моденов - Дифференциальные уравнения (курс лекций), страница 3

Îáîçíà÷àÿ ðàçíîñòü ýòèõ ðåøåíèéy(x, µ + ∆µ) − y(x, µ) = U (x, µ, ∆µ),17ïîëó÷èì:dU= [f (x, y(x, µ + ∆µ), µ + ∆µ) − f (x, y(x, µ), µ + ∆µ)] +|{z}dx∆1 (x)+ [f (x, y(x, µ), µ + ∆µ) − f (x, y(x, µ), µ)] = ∆1 (x) + ∆2 (x),|{z}∆2 (x) U (x ) = 00⇔ U=ïî Ë0.1Zx∆1 (ξ)dξ +Zx(*)∆2 (ξ)dξx0x0 ñèëó óñëîâèÿ Ëèïøèöà 2)|∆1 (x)| 6 L |U (x, µ, ∆µ)|Èç óñëîâèÿ 1) òåîðåìû óíêöèÿ f (x, y, µ) - íåïðåðûâíà ïî µ â Π - çàìêíóòîéîãðàíè÷åííîé îáëàñòè.

Çíà÷èò, ïî òåîðåìå Êàíòîðà f (x, y(x, µ), µ) - ðàâíîìåðíîíåïðåðûâíà ïî µ ⇔ ∀ε > 0, ∃δ(ε) > 0: |∆µ| < δ(ε), ⇒ |∆2 (x)| < Hε ñðàçó äëÿdef∀x ∈ [x0 , x0 + H]. Ñëåäîâàòåëüíî, ïîëàãàÿ Z = |U | = |U (x, µ, ∆µ)|, èç (*) èìååìíåðàâåíñòâî ëåììû ðîíóîëëà ñ êîíñòàíòîé C = ε:(∗) ⇒ Z(x, µ, ∆µ) < ε + LZxZ(ξ, µ, ∆µ)dξ⇒ë.

ðîíóîëëàZ(x, µ, ∆µ) < εeLH .x0Èòàê, ∀ε > 0, ∃δ(ε) > 0 òàêîå, ÷òî åñëè |∆µ| < δ(ε), òî|y(x, µ + ∆µ) − y(x, µ)| < εeLH , ∀x ∈ [x0 , x0 + H]⇔ y = y(x, µ) - íåïðåðûâíàÿ ïî µ óíêöèÿ ∀µ ∈ [µ0 − c, µ0 + c].def1.3Ñóùåñòâîâàíèåè åäèíñòâåííîñòü ðåøåíèÿ çàäà÷è Êîøè äëÿíîðìàëüíîé ñèñòåìû óðàâíåíèé: dyi = fi (x, y1 , ..., yn )dxyi (x0 ) = yi0 ,Òåîðåìà 1.3. (Áåç äîêàçàòåëüñòâà)©Ïóñòü â îáëàñòè D =|x − x0 | 6 a,fi (x, y1 , ..., yn ):1)íåïðåðûâíûåïîñîâîêóïíîñòè18(1.5)i = 1, n¯¯ª¯yi − y 0 ¯ 6 bi îïðåäåëåíû óíêöèèiàðãóìåíòîââîáëàñòèD,³´∃M = max max|fi (x, y1 , ..., yn )|iD2) óäîâëåòâîðÿþùèå â îáëàñòè D ïîyi óñëîâèþ ËèïøèöàrïåðåìåííûìnP(ȳi − ȳ¯i )2 , L > 0.|fi (x, y¯1 , ..., y¯n ) − fi (x, y¯1 , ..., y¯n )| 6 Li=1Òîãäà çàäà÷à Êîøè (1.5) ¡èìååò¢ åäèíñòâåííîå ðåøåíèå yi íà ïðîìåæóòêåbi.∀x ∈ [x0 , x0 + H], ãäå H = min a, MiÇàìå÷àíèå.Âåêòîðíàÿ çàïèñü: ~y = {y1 , ..., yn }. d~y = f~ (x, ~y ),dxÇàäà÷à Êîøè:.~y (x¯ 0 ) = ~y0 .¯¯¯Óñëîâèå Ëèïøèöà: ¯f~ (x, ~y1 ) − f~ (x, ~y2 )¯ 6 L |~y1 − ~y2 |, L>0.1.4Ïðîñòåéøèå äèåðåíöèàëüíûå óðàâíåíèÿ 1ãî ïîðÿäêà, èíòåãðèðóåìûå â êâàäðàòóðàõ.1.4.1.

Óðàâíåíèå ñ ðàçäåëÿþùèìèñÿ ïåðåìåííûìèg(y) = 0g(y) = 0dyyx= f (x)g(y) ⇐⇒  dy = f (x)dx ⇐⇒  R dη = R f (ξ)dξdxg(y)x0y0 g(η)4.2.Ëèíåéíîå îäíîðîäíîå óðàâíåíèå.Y (x) = 0dY= p(x)Y (x) ⇔ dY = p(x)dx ⇔dxYY (x) = 0Y (x) = 0RxRx⇔⇔p(ξ)dξ ⇔ln |Y | = p(ξ)dξ + ln C̄, C̄ > 0x0|Y | = C̄ex0Y (x) = 0RxRxp(ξ)dξp(ξ)dξ, ãäå Ñ - ëþáàÿ êîíñòàíòà.⇔ Y (x) = Cex0⇔  Y = C̄ex0Rxp(ξ)dξY = −C̄ex0Rxp(ξ)dξÈòàê,= p(x)Y (x) ⇔ Y (x) = Ce.1.4.3. Ëèíåéíîå íåîäíîðîäíîå óðàâíåíèå.

Ìåòîä âàðèàöèè ïîñòîÿííîé.àññìîòðèìdYdxx0y ′ (x) = p(x)y(x) + f (x).19(1.6)Èùåì ðåøåíèå â âèäåRxp(ξ)dξ(1.7)y(x) = C(x)ex0Èç óðàâíåíèÿ (1.6) èìååì:Rx′Rxp(ξ)dξC (x)ex0−p(ξ)dξ+ f (x) ⇔= p(x)C(x)ex0+ C(x)p(x)ex0RxRxp(ξ)dξp(ξ)dξ⇔ C ′ (x) = f (x)e x0RηZx− p(ξ)dξdη⇔ C(x) = C + f (η)e x0⇔ C(x) = C +x0ZxxR0p(ξ)dξf (η)e η(1.8)dηx0Ïîäñòàâëÿÿ (1.8) â (1.7), ïîëó÷èìRxy(x) = Cex0p(ξ)dξ+Zxx0RxÅñëè K(x, η) = eη′p(ξ)dξxR0p(ξ)dξ+f (η) e ηRxp(ξ)dξ dη.x0|{z}K(x,η)- èìïóëüñíàÿ óíêöèÿ , òî:Rxp(ξ)dξy (x) = p(x)y(x) + f (x) ⇔ y(x) = |Ce {zx0Y (x)}+ZxK(x, η)f (η)dη = Y (x) + ȳ(x),x0|{zȳ(x)}ãäå Y (x) - îáùåå ðåøåíèå îäíîðîäíîãî óðàâíåíèÿ, ȳ(x) - ÷àñòíîå ðåøåíèåíåîäíîðîäíîãî.1.5Ïîíÿòèå î "êîððåêòíî ïîñòàâëåííûõ"çàäà÷àõ.Äîïîëíèòåëüíûå óñëîâèÿ (íà÷àëüíûå, ãðàíè÷íûå è ò.ä.), îáåñïå÷èâàþùèåìàòåìàòè÷åñêîé çàäà÷å èçè÷åñêóþ îïðåäåëåííîñòü, íàçûâàþò äàííûìèçàäà÷è .Ìàòåìàòè÷åñêàÿ çàäà÷à, óäîâëåòâîðÿþùàÿ òðåì òðåáîâàíèÿì:1.

ðåøåíèå åå äîëæíî ñóùåñòâîâàòü,2. ðåøåíèå äîëæíî áûòü åäèíñòâåííûì,3. ðåøåíèå äîëæíî íåïðåðûâíî çàâèñåòü îò äàííûõ çàäà÷è (ò.å. áûòü óñòîé÷èâûì)- ìàëûì èçìåíåíèÿì ëþáûõ äàííûõ çàäà÷è äîëæíû ñîîòâåòñòâîâàòü ìàëûåèçìåíåíèÿ ðåøåíèÿ,íàçûâàåòñÿ "êîððåêòíî"ïîñòàâëåííîé .20Çàìå÷àíèå. Ñóùåñòâîâàíèå è åäèíñòâåííîñòü ðåøåíèÿ çàäà÷è Êîøè äëÿóðàâíåíèÿ n-ãî ïîðÿäêà, ðàçðåøåííîãî îòíîñèòåëüíî ñòàðøåé ïðîèçâîäíîé:¡¢ (n)′(n−1)y(x)=fx,y,y,...,y y(x0 ) = y0y ′ (x0 ) = y0′y = y(x) :··············· (n−1)(n−1)y(x0 ) = y0ñâîäèòñÿ ê çàäà÷å Êîøè äëÿ íîðìàëüíîé ñèñòåìû 1 -ãî ïîðÿäêà.

Ïîëîæèìy ′ (x) = y1 (x)y ′′ (x) = y2 (x)···············y (n−1) (x) = yn (x)Ïîëó÷èì çàäà÷ó Êîøè:yi (x) :1.6Ïðèìåðdy= y1 (x),dxdy1= y2 (x),dx···············.dyn−1= yn (x),dxdyn= f (x, y, y1 , y2 , ..., yn ),dxyi (x0 ) = yi0 , i = 1, n.×èñëåííàÿ ðåàëèçàöèÿ ìåòîäà Ýéëåðà.½ ′ xyx2y = 2 , x ∈ [0, 1]⇐⇒ y = e 4y(0) = 1h = 0.121kxkyk01234567891000.10.20.30.40.50.60.70.80.91.0111.0051.01511.03031.05091.07721.10951.14831.19421.2479f (xk , yk ) =x k yk2∆yk = 0.1f (xk , yk )00.050.10050.15230.20670.26270.32320.38830.45930.537400.0050.01010.01520.02060.02630.03230.03880.04590.0537”òî÷íîå çíà÷åíèå”x2y=e411.00251.01001.02271.04081.06451.09421.13031.17351.22441.28401.

Àáñîëþòíàÿ ïîãðåøíîñòü çíà÷åíèÿ y10 ñîñòàâëÿåò ε10 = 1, 2840 − 1, 2479 =0, 0361.2. Îòíîñèòåëüíàÿ ïîãðåøíîñòü ≈ 3%.22ëàâà 2Ëèíåéíûå óðàâíåíèÿ2.1nãîïîðÿäêà.Ñâîéñòâà ëèíåéíûõ óðàâíåíèé.Îïðåäåëåíèå 2.1. Ëèíåéíîå óðàâíåíèå nãî ïîðÿäêà ⇔DefLy = y (n) + a1 (x)y (n−1) + ... + an (x)y = f (x),(2.1)ãäå y = y(x), x ∈ [a, b].Îïðåäåëåíèå 2.2. Åñëè f (x) ≡ 0, òî ëèíåéíîå óðàâíåíèå(2.1′ )Ly = 0íàçûâàþò îäíîðîäíûì.Îïðåäåëåíèå 2.3. Óñëîâèÿ, çàäàþùèå óíêöèþ y(x) è å¼ ïðîèçâîäíûå ïðè x0 ∈[a, b]:(n−1)y (x0 ) = y0 , y ′ (x0 ) = y0′ , ..., y (n−1) (x0 ) = y0,(2.2)íàçûâàþò íà÷àëüíûìè.Îïðåäåëåíèå 2.4. Çàäà÷ó (2.1), (2.2) íàçûâàþò çàäà÷åé Êîøè.Óòâåðæäåíèå.

(Òåîðåìà ñóùåñòâîâàíèÿ è åäèíñòâåííîñòè ðåøåíèÿçàäà÷è Êîøè (2.1), (2.2)) Äëÿ äèåðåíöèàëüíîãî óðàâíåíèÿ (2.1) ñíåïðåðûâíûìè êîýèöèåíòàìè è ïðàâîé ÷àñòüþ íà [a,b℄ ñóùåñòâóåòåäèíñòâåííîå ðåøåíèå çàäà÷è Êîøè (2.1), (2.2).Òåîðåìà2.1 (Ïðèíöèïñóïåðïîçèöèè). Ïóñòü:ª©1) yk (x) , k = 1, N :Lyk = fk (x)NP2) Ly = f (x) =Ck fk (x), (Ck - çàäàííûå ïîñòîÿííûå)k=1Òîãäà: y =NPCk yk (x), (Ck - òå æå ïîñòîÿííûå).k=1ÄîêàçàòåëüñòâîLy = LNXk=1Ck yk (x)=L - ëèí.

îï.NXCk Lyk (x) =1)k=123NXk=1Ck fk (x) = f (x), ÷.ò.ä.2)Ñëåäñòâèå 2.1.©ªyk (x) , k = 1, N : Lyk = 0 ⇒ y =NXCk yk (x) : Ly = 0,k=1∀Ck .Äîêàçàòåëüñòâî Ïîëîæèòü â Ò.2.1 fk (x) ≡ 0, ∀k = 1, N .2.2Ëèíåéíîå îäíîðîäíîå óðàâíåíèå.Íàïîìíèì îïðåäåëåíèÿ è òåîðåìó, äîêàçàííóþ â ìàò. àíàëèçå.ª©Îïðåäåëåíèå 2.5. Ôóíêöèè yk (x), k = 1, n ëèíåéíî çàâèñèìû ∀x ∈ [a, b] ⇔Def∃Ck - êîíñòàíòû,nPk=1|Ck | =6 0 (íå âñå ðàâíû íóëþ):C1 y1 (x) + C2 y2 (x) + ...

+ Cn yn (x) ≡ 0, ∀x ∈ [a, b](*)©ªÎïðåäåëåíèå 2.6. Ôóíêöèè yk (x), k = 1, n ëèíåéíî íåçàâèñèìû ∀x ∈[a, b] ⇔ (∗) ⇒ 1 = 2 = ... = n = 0.DefÎïðåäåëåíèå 2.7. Îïðåäåëèòåëü Âðîíñêîãî ⇔Def¯¯ y1 (x)y2 (x)¯ ′¯ y1 (x)y2′ (x)W (x) ≡ W [y1 (x), y2 (x), ..., yn (x)] ≡ ¯¯......¯ (n−1)(n−1)¯y(x)(x) y21¯...yn (x) ¯¯...yn′ (x) ¯¯...... ¯¯(n−1)(x)¯... ynÒåîðåìà2.2 ª(Íåîáõîäèìîå óñëîâèå ëèíåéíîé çàâèñèìîñòè:).

Ôóíêöèè©yk (x), k = 1, n ëèíåéíî çàâèñèìû ∀x ∈ [a, b] ⇒ W (x) ≡ 0, ∀x ∈ [a, b].Ñëåäñòâèå 2.2. (Îáðàòíàÿ òåîðåìà - äîñòàòî÷íîåóñëîâèå ªëèíåéíîé©íåçàâèñèìîñòè): ∃x0 ∈ [a, b], W (x0 ) 6= 0 ⇒ óíêöèè yk (x), k = 1, n ëèíåéíîíåçàâèñèìû ∀x ∈ [a, b].Òåîðåìà2.3. ÎïðåäåëèòåëüÂðîíñêîãîëèíåéíîíåçàâèñèìûõðåøåíèé îäíîðîäíîãî äèåðåíöèàëüíîãî óðàâíåíèÿ n ïîðÿäêà ñ íåïðåðûâíûìèêîýèöèåíòàìè (2.1′ ) îòëè÷åí îò íóëÿ.Äîêàçàòåëüñòâî (îò ïðîòèâíîãî).Ïðåäïîëîæèì, ÷òî ∃x0 ∈ [a, b] : W (x0 ) = 0.

Ñîñòàâèì îäíîðîäíóþ ÑËÀÓñ îïðåäåëèòåëåì W (x0 ) = 0:C1 y1 (x0 ) + ... + Cn yn (x0 ) = 0,C1 y1′ (x0 ) + ... + Cn yn′ (x0 ) = 0,(2.3)......................................................(n−1)(n−1)C1 y1(x0 ) + ... + Cn yn(x0 ) = 0.24Ýòà ñèñòåìà èìååò íåòðèâèàëüíîå ðåøåíèå:à n!X¡ 0 0¢~ 0 6= ~0,∃ C , C , ..., C 0 = C|C 0 | =6 0 .12nkk=1àññìîòðèì óíêöèþ(2.4)y = C10 y1 (x) + C20 y2 (x) + ... + Cn0 yn (x) ñèëó Ñë.2.1 Ò.2.1 (ïðèíöèïà ñóïåðïîçèöèè) è (2.3) óíêöèÿ (2.4) ÿâëÿåòñÿðåøåíèåì çàäà÷è Êîøè (äëÿ îäíîðîäíîãî óðàâíåíèÿ ñ íóëåâûìè íà÷àëüíûìèóñëîâèÿìè):¾Ly = 0,y(x0 ) = 0, y ′ (x0 ) = 0, ..., y (n−1) (x0 ) = 0è, ñëåäîâàòåëüíî, òîæäåñòâåííî ðàâíà íóëþ:y = C10 y1 (x) + C20 y2 (x) + ...

+ Cn0 yn (x) ≡ 0, ∀x ∈ [a, b]©ª⇔ yk (x), k = 1, n ëèíåéíî çàâèñèìû ∀x ∈ [a, b], ÷òî ïðîòèâîðå÷èò óñëîâèþDef 2.51) òåîðåìû. Ñëåäîâàòåëüíî, W (x) 6= 0, ∀x ∈ [a, b], ÷.ò.ä.Ñâîéñòâà îïðåäåëèòåëÿ Âðîíñêîãîóðàâíåíèÿ Lyk©= 0, (k = 1, n)∀xª ∈ [a, b]:èçðåøåíèéîäíîðîäíîãî10 W (x) = 0 ⇔ ©yk (x), k = 1, nª ëèíåéíî çàâèñèìû.20 W (x) 6= 0 ⇔ yk (x), k = 1, n ëèíåéíî íåçàâèñèìû.Îïðåäåëåíèå 2.8. Ëèíåéíî íåçàâèñèìûå ðåøåíèÿ îäíîðîäíîãî óðàâíåíèÿ (2.1′ )ñ íåïðåðûâíûìè êîýèöèåíòàìè íàçûâàþò óíäàìåíòàëüíîé ñèñòåìîéðåøåíèé (ÔÑ) ýòîãî óðàâíåíèÿ.Ñëåäñòâèå 2.3.

Îïðåäåëèòåëü Âðîíñêîãî èç ÔÑ îòëè÷åí îò íóëÿ.Òåîðåìà 2.4 (î ñóùåñòâîâàíèè ÔÑ). Âñÿêîå ëèíåéíîå îäíîðîäíîå óðàâíåíèåñ íåïðåðûâíûìè êîýèöèåíòàìè (2.1′ ) èìååò ÔÑ.Äîêàçàòåëüñòâî . Çàäàäèì ïðîèçâîëüíûé ÷èñëîâîé, îòëè÷íûé îò íóëÿ,îïðåäåëèòåëü :¯¯ a11¯¯a∆0 = ¯¯ 21¯ ...¯an1... a1k... a2k... ...... ankÏîñòðîèì n ðåøåíèé çàäà÷è Êîøè:Lyk (x) = 0,y k (x0 ) = a1k ,yk′ (x0 ) = a2k ,....................... (n−1)yk(x0 ) = ank .25¯...

a1n ¯¯... a2n ¯¯6 0.=... ... ¯¯... ann ¯ÀëãîðèòìïîñòðîåíèÿÔÑ©ªÏîêàæåì , ÷òî ýòè ðåøåíèÿ yk (x), k = 1, n îáðàçóþò ÔÑ. Ñîñòàâèìîïðåäåëèòåëü Âðîíñêîãî èç©ðåøåíèé yk (x)ª â ò. x0 :äîñò. óñë. ë. íåçàâ.W (x0 ) = ∆0 6= 0⇒yk (x), k = 1, n ëèíåéíî íåçàâèñèìû ∀x ∈ [a, b]Ñë2.2Îïð.2.8©ª⇒yk (x), k = 1, n - ÔÑ óðàâíåíèÿ 2.1′ , ÷.ò.ä.Çàìå÷àíèå. Òàê êàê îïðåäåëèòåëåé âèäà ∆0 ñêîëüêî óãîäíî, òî ñóùåñòâóåòáåñêîíå÷íîå ÷èñëî ÔÑ äëÿ óðàâíåíèÿ Ly = 0.Òåîðåìà 2.5. (î ïîñòðîåíèè îáùåãî ðåøåíèÿ ëèíåéíîãîîäíîðîäíîãäèåðåíöèàëüíîãî óðàâíåíèÿ n ïîðÿäêà).

Ïóñòü yk (x), k = 1, n - ÔÑóðàâíåíÿ 2.1′ .Òîãäà ëþáîå ðåøåíèå Z ýòîãî óðàâíåíèÿ ïðåäñòàâèìî â âèäå:Z=nXCk yk (x),k=1ãäå Ck - íåêîòîðûå ïîñòîÿííûå.Äîêàçàòåëüñòâî Ïóñòü Z = Z(x) - êàêîå-ëèáî ðåøåíèå çàäà÷è Êîøè:LZ(x) = 0,Z(x0 ) = Z0 ,′′Z (x0 ) = Z0 ,(**)................(n−1) Z (n−1) (x0 ) = Z0.©ªÏîêàæåì , ÷òî ∃ Ck , k = 1, n : C1 y1 (x) + C2 y2 (x) + ... + Cn yn (x) ≡ Z, ∀x ∈ [a, b]è ïðè x = x0 ∈ [a, b] âûïîëíÿþòñÿ íà÷àëüíûå óñëîâèÿC1 y1 (x0 ) + C2 y2 (x0 ) + ... + Cn yn (x0 ) = Z0 ,ÑËÀÓ......................................................................ñ îïðåäåëèòåëåì(n−1) (n−1)(n−1)(n−1)W (x0 )(x0 ) = Z0C1 y1(x0 ) + C2 y2(x0 ) + ...