В.П. Моденов - Дифференциальные уравнения (курс лекций), страница 7

Ïî îðìóëàì Êðàìåðà íàõîäèì ýòîðåøåíèå:¯¯¯¯¯ y1 0 ¯¯ 0 y2 ¯¯ = y1 f1 ,¯ = −y2 f1 ,∆2 = ¯¯ ′∆1 = ¯¯y1 f1 ¯f1 y2′ ¯C1′ = −y2 (x)f (x),p(x)W (x)C2′ =y1 (x)f (x).p(x)W (x)Ïî îðìóëå Ëèóâèëëÿ-Îñòðîãðàäñêîãî p(x)W (x) = C . Ñëåäîâàòåëüíî,1C1′ = − y2 (x)f (x),1C2′ = y1 (x)f (x).Èíòåãðèðóÿ, áóäåì èìåòü:1C2 (x) =CZxy1 (s)f (s)ds + C20 ,01C1 (x) = −CZxy2 (s)f (s)ds +C10l481=CZxly2 (s)f (s)ds + C10 .Ïîäñòàâëÿÿ C1 (x) è C2 (x) â èñêîìûé âèä ðåøåíèÿ (4.3), ïîëó÷èì:y1 (x)y(x) =C+Zlx0C1 y1 (x)y2 (x)y2 (s)f (s)ds +CZxy1 (s)f (s)ds+0+C20 y2 (x).Óäîâëåòâîðÿÿ ãðàíè÷íûì óñëîâèÿì (à) è (á), ó÷èòûâàÿ 10 , íàéäåì êîíñòàíòûèíòåãðèðîâàíèÿ:C20 = 0,C10 = 0.Ñëåäîâàòåëüíî,y(x) =Zxy1 (s)y2 (x)f (s)ds +C0Zly2 (s)y1 (x)f (s)ds.CxÎáîçíà÷èì:1G(s, x) =C(y1 (s)y2 (x),y2 (s)y1 (x),åñëè 0 6 s 6 x,åñëè x 6 s 6 l.G(s, x) - óíêöèÿ ðèíà.

Èç âèäà óíêöèè ðèíà ñëåäóåò, ÷òî îíà ñèììåòðè÷íàÿ:(1 y1 (x)y2 (s), åñëè 0 6 x 6 s,G(x, s) =(***)C y2 (x)y1 (s), åñëè s 6 x 6 l.Ñëåäîâàòåëüíî,y(x) =ZlG(x, s)f (s)ds.(4.4)0Òàêèì îáðàçîì, äîêàçàíà ñëåäóþùàÿÒåîðåìà 4.1. Åñëè îäíîðîäíàÿ êðàåâàÿ çàäà÷à (4.1′ ) èìååò òîëüêî òðèâèàëüíîåðåøåíèå, òî ñóùåñòâóåò åäèíñòâåííîå ðåøåíèå íåîäíîðîäíîé êðàåâîé çàäà÷è(4.1) äëÿ ëþáîé íåïðåðûâíîé íà îòðåçêå [0, l] óíêöèè f (x) è äàåòñÿ îðìóëîé(4.4).4.3Ñâîéñòâà óíêöèè ðèíà (***).10 G(x, s) = z(x) óäîâëåòâîðÿåò îäíîðîäíîìó óðàâíåíèþ (4.1′ ) L[z] = 0 ïðè x 6= s.20 G(x, s) = z(x) óäîâëåòâîðÿåò íóëåâûì ãðàíè÷íûì óñëîâèÿìz(0) = G(0, s) = 0, ò.ê.

y1 (0) = 0,z(l) = G(l, s) = 0, ò.ê. y2 (l) = 0.4930 Íåïðåðûâíà: G(x, s) = z(x) ∈ C[0, l], ò.ê. z(s − 0) = z(s + 0) =40 ż ïåðâàÿ ïðîèçâîäíàÿ èìååò ïðè x = s ðàçðûâ I ðîäà:¯x=s+0¯x=s+0dz ¯¯dG(x, s) ¯¯1.==¯¯dx x=s−0dxp(s)x=s−01y (s)y2 (s).C 1 ñàìîì äåëå, (***) =⇒¯¯dG(x, s) ¯¯1dG(x, s) ¯¯1−= y2′ (s)y1 (s) − y1′ (s)y2 (s) =¯¯dxdxCCx=s+0x=s−01W (s)=, ÷.ò.ä.=.

Ëèóâèëëÿ-Îñòð. p(s)W (s)p(s)Çàìå÷àíèå. Ñïðàâåäëèâî îáðàòíîå óòâåðæäåíèå.Ôóíêöèÿ, óäîâëåòâîðÿþùàÿ óñëîâèÿì 10 − 40 , èìååò âèä (***), ò.å. ÿâëÿåòñÿóíêöèåé ðèíà.Îòñþäà, âòîðîé ñïîñîá íàõîæäåíèÿ óíêöèè ðèíà.Èùåì åå â âèäå(C1 (s)y1 (x), åñëè 0 6 x 6 s,G(x, s) =C2 (s)y2 (x), åñëè s 6 x 6 l,ãäå C1 (s) è C2 (s) íàõîäÿòñÿ èç óñëîâèé 30 è 40 .Ôèçè÷åñêèé ñìûñë óíêöèè ðèíà - óíêöèè èñòî÷íèêà.àññìîòðèì êðàåâóþ çàäà÷ó:L[y] = δ(x − x0 )y(0) = 0,y(l) = 0, x0 ∈ (0, l).åøåíèå ìåòîäîì óíêöèè ðèíà äàåò:y(x) =ZlG(x, s)δ(s − x0 )ds = G(x, x0 ),0ò.å. G(x, x0 ) - ýòî çíà÷åíèå y(x) â ò. x, åñëè â îêðåñòíîñòè òî÷êè ò.

x0 ñîñðåäîòî÷åíèñòî÷íèê f (x) = δ(x − x0 ).4.4Ïðèìåð. (Ñòàòè÷åñêàÿ çàäà÷à î ïðîèëåñòðóíû.)′′y (x) = f (x)y(0) = 0,y(1) = 0.(p = 1, q = 0, f (x) ∈ C[0, 1])50åøåíèå.1). àññìîòðèì îäíîðîäíóþ çàäà÷ó:′′y (x) = 0,y(0) = 0,y(1) = 0,⇐⇒ y ≡ 0.Îäíîðîäíàÿ çàäà÷à èìååò òîëüêî íóëåâîå ðåøåíèå, ñëåäîâàòåëüíî, èñõîäíàÿçàäà÷à èìååò åäèíñòâåííîå ðåøåíèå.2). Ïîñòðîèì óíêöèþ ðèíà. àññìîòðèì ðåøåíèÿ äâóõ çàäà÷ Êîøè(ny1′′ (x) = 0.=⇒ y1 (x) = x;y1 (x) :y1 (0) = 0(y2′′ (x) = 0,y2 (x) :y2 (1) = 0n.=⇒ y2 (x) = x − 1.1é ñïîñîá.1(∗ ∗ ∗) =⇒ G(x, s) =Cãäå(x(s − 1),(x − 1)s,åñëè 0 6 x 6 s,åñëè s 6 x 6 1,¯¯¯x x − 1¯¯¯ = 1.C = p(x)W (x) = ¯11 ¯Ñëåäîâàòåëüíî,2é ñïîñîá.

Èùåì(x(s − 1), 0 6 x 6 s,.G(x, s) =(x − 1)s, s 6 x 6 1.(C1 (s)y1 (x) = C1 (s)x,0 6 x 6 s,.G(x, s) =C2 (s)y2 (x) = C2 (s)(x − 1), s 6 x 6 1Óäîâëåòâîðÿåì óñëîâèÿì 30 è 40 :(C1 (s)s = C2 (s)(s − 1)C2 (s) − C1 (s) = 13). åøåíèå çàäà÷è: y(x) =R1(C1 (s) = (s − 1),=⇒C2 (s) = s.G(x, s)f (s)ds. Ôèçè÷åñêèé ñìûñë - ïðîèëü ñòðóíû0ïðè íàãðóçêå f (x). ÷àñòíîñòè,y ′′ (x) = δ(x − 32 ),y(0) = 0,y(1) = 0.µ2⇐⇒ y(x) = G x,351¶=(− 1 x,= 2 3(x − 1),34.5åñëè 0 6 x 6 23 ,åñëè 23 6 x 6 1.Çàìå÷àíèÿ.Çàìå÷àíèå 1.

Ïðîèçâîëüíîå äèåðåíöèàëüíîå óðàâíåíèå âòîðîãî ïîðÿäêà:a(x) y ′′ (x) + b(x)y ′ (x) + (x)y(x) = g(x)a(x)6=0ìîæåò áûòü ïðèâåäåíî ê âèäó (4.1). Óìíîæèì íà0, p′ (x) =Ïîëó÷èìb(x)p(x).a(x)p(x)y ′′ (x) +p(x)a(x)R6= 0, ãäå p(x) = eb(x)dxa(x)>b(x)b(x) R a(x)(x)p(x)dx ′y (x) + p(x)y(x) =eg(x)a(x)a(x)a(x){z}| {z }| {z }|p′ (x)−q(x)f (x)·¸dydp(x)− q(x)y = f (x), ÷.ò.ä.⇐⇒ L[y] =dxdxÇàìå÷àíèå 2. Çàäà÷à ñ íåîäíîðîäíûìè ãðàíè÷íûìè óñëîâèÿìè ìîæåò áûòüñâåäåíà ê çàäà÷å ñ íóëåâûìè ãðàíè÷íûìè óñëîâèÿìè.L[y] = g(x),y 2 − y1 xy(x) : y(0) = y1 ,Z(x) = y(x) − y+1:l|{z}y(l) = y .2ãäåA+BxL[Z] = g(x) − L[A + Bx] = f (x),Z(0) = 0,Z(l) = 0,A = y1 ,1B = y2 −y.lÇàìå÷àíèå 3. Àíàëîãè÷íî ðåøàþòñÿ êðàåâûå çàäà÷è ñ ãðàíè÷íûìè óñëîâèÿìèII è III ðîäà.52ëàâà 5Óðàâíåíèÿ â ÷àñòíûõ ïðîèçâîäíûõïåðâîãî ïîðÿäêà.5.1Ëèíåéíîå îäíîðîäíîå óðàâíåíèå.1.

Îïðåäåëåíèÿ. Ëèíåéíûì îäíîðîäíûì äèåðåíöèàëüíûì óðàâíåíèåì â÷àñòíûõ ïðîèçâîäíûõ íàçûâàþò óðàâíåíèÿ âèäànX∂z∂z∂z∂zXi= X1 (x1 , x2 , · · · , xn )+ X2 (· · · )+ · · · + Xn (· · · )= 0,∂x∂x∂x∂xi12ni=1(5.1)ãäå Xi (x1 , · · · , xn ) - êîýèöèåíòû óðàâíåíèÿ, îïðåäåëåíû è äèåðåíöèðóåìûâ îáëàñòè D ïåðåìåííûõ x1 , · · · , xn , è îäíîâðåìåííî íå îáðàùàþòñÿ â íóëü íè ânPXi2 6= 0 â D.îäíîé òî÷êå D:i=1åøåíèåì óðàâíåíèÿ (5.1) íàçûâàþò óíêöèþ z = z(x1 , x2 , · · · , xn ),îáëàäàþùóþ ÷àñòíûìè ïðîèçâîäíûìè è îáðàùàþùóþ (5.1) â òîæäåñòâî.2. Óðàâíåíèÿ õàðàêòåðèñòèê.Îïðåäåëåíèå 5.1. Õàðàêòåðèñòè÷åñêîé ñèñòåìîé, ñîîòâåòñòâóþùåé(5.1), íàçûâàþò ñèñòåìó îáûêíîâåííûõ äèåðåíöèàëüíûõ óðàâíåíèédx1dx2dxn== ··· =.X1 (x1 , x2 , · · · , xn )X2 (· · · )Xn (· · · )Îïðåäåëåíèå 5.2.(5.2).(5.2)Õàðàêòåðèñòèêè óðàâíåíèÿ (5.1) - ðåøåíèÿ ñèñòåìûÑ÷èòàåì, ÷òî ÷åðåç êàæäóþ òî÷êó îáëàñòè D ìîæåò ïðîõîäèòü åäèíñòâåííàÿõàðàêòåðèñòèêà.Çàìå÷àíèå Ïóñòü Xn6= 0 â D.

Òîãäà ìîæíî ñ÷èòàòü xn íåçàâèñèìîéïåðåìåííîé (ïàðàìåòðîì). Õàðàêòåðèñòè÷åñêàÿ ñèñòåìà óðàâíåíèé (5.2)ïðèíèìàåò âèäx1 = x1 (xn ),·········óð-ÿ õàð-ê (5.2')dx1X1dxn−1Xn−1=,··· ,=⇐⇒dxnXndxnXnxn−1 = xn−1 (xn ),xn = xn - ïàðàìåòð533. Ïåðâûå èíòåãðàëû.Îïðåäåëåíèå 5.3. Ψ(x1 , x2 , · · · , xn ) - ïåðâûé èíòåãðàë (5.2) ⇐⇒Def⇐⇒ Ψ(x1 (xn ), x2 (xn ), · · · , xn−1 (xn ), xn ) ≡ C .DefÒåîðåìà 5.1. Âñÿêîå ðåøåíèå îäíîðîäíîãî óðàâíåíèÿ (5.1) ÿâëÿåòñÿ ïåðâûìèíòåãðàëîì ñèñòåìû (5.2) è îáðàòíî: âñÿêèé ïåðâûé èíòåãðàë (5.2) ÿâëÿåòñÿðåøåíèåì (5.1).Äîêàçàòåëüñòâî.1). Ïóñòü z = Ψ(x1 , x2 , · · · , xn ) - ðåøåíèå óðàâíåíèÿ (5.1).

Òîãäà ïî îïðåäåëåíèþ(5.2)∂Ψ∂Ψ+ · · · + Xn≡ 0, â D.X1(5.3)∂x1∂xnÑ÷èòàÿ Xn 6= 0 â D, à xn - íåçàâèñèìîé ïåðåìåííîé, çàìåíèì óðàâíåíèå (5.3)ðàâíîñèëüíûìX1 ∂ΨXn−1 ∂Ψ∂Ψ+ ··· ++≡0âDXn ∂x1Xn ∂xn−1 ∂xnèëè, â ñèëó (5.2′ ) óðàâíåíèåì∂Ψ dxn−1∂Ψ∂Ψ dx1+ ··· ++≡ 0 â D.∂x1 dxn∂xn−1 dxn∂xn(5.4)Ïðèìåíÿÿ ïðàâèëî äèåðåíöèðîâàíèÿ ñëîæíîé óíêöèè, ïîëó÷èìdΨ (x1 (xn ), x2 (xn ), · · · xn−1 (xn ), xn ) ≡ 0 â D.{z}dxn |õàðàêòåðèñòèêàÑëåäîâàòåëüíî, ïî îïðåäåëåíèþ 5.3Ψ(x1 , x2 , · · · , xn ) - ïåðâûé èíòåãðàë (5.2).2). Ïóñòü Ψ(x1 , x2 , · · · , xn ) - ïåðâûé èíòåãðàë (5.2).

Òîãäà ïî îïðåäåëåíèþ 5.3Ψ(x1 (xn ), x2 (xn ), · · · , xn−1 (xn ), xn ) ≡ C∀xn ∈ D.∂Ψ dx1∂Ψ dxn−1∂Ψ+ ··· ++≡ 0,∂x1 dxn∂xn−1 dxn∂xn∀xn ∈ D.Äèåðåíöèðóÿ ïî xn , ïîëó÷èì(5.4)Ñëåäîâàòåëüíî, â ñèëó (5.2′ ) âäîëü õàðàêòåðèñòèêèX1 ∂ΨXn−1 ∂Ψ∂Ψ+ ··· ++≡ 0.Xn ∂x1Xn ∂xn−1 ∂xnÍî, ÷åðåç êàæäóþ òî÷êó D ïðîõîäèò åäèíñòâåííàÿ õàðàêòåðèñòèêà, ïîýòîìóâûïîëíÿåòñÿ òîæäåñòâåííî ðàâåíñòâîX1∂Ψ∂Ψ+ · · · + Xn≡ 0â D.∂x1∂xnÝòî îçíà÷àåò, ÷òî ïî îïðåäåëåíèþ Ψ(x1 , x2 , · · · , xn ) - ðåøåíèå (5.1), ÷.ò.ä.54(5.3)4.

Îáùåå ðåøåíèå.Òåîðåìà 5.2. Ïóñòü:10 . Èçâåñòíû (n-1) ïåðâûõ èíòåãðàëîâ õàðàêòåðèñòè÷åñêîé ñèñòåìû (5.2):Ψ1 (x1 , · · · , xn−1 , xn ) ≡ C1···························Ψn−1 (x1 , · · · , xn−1 , xn ) ≡ Cn−120 . Îíè íåçàâèñèìû, ò.å. ÿêîáèàíD (Ψ1 , · · · , Ψn−1 )6= 0 â D.D(x1 , · · · , xn−1 )Òîãäà z = Φ (Ψ1 , · · · , Ψn−1 ), ãäå Φ - ïðîèçâîëüíàÿ äèåðåíöèðóåìàÿ óíêöèÿ,ÿâëÿåòñÿ îáùèì ðåøåíèåì (5.1).Îáùåå ðåøåíèå óðàâíåíèÿ (5.1) åñòü ïðîèçâîëüíàÿ äèåðåíöèðóåìàÿóíêöèÿ (n − 1) íåçàâèñèìûõ ïåðâûõ èíòåãðàëîâ.Äîêàçàòåëüñòâî 1). Äîêàæåì, ÷òî äèåðåíöèðóåìàÿ óíêöèÿΦ (Ψ1 , · · · , Ψn−1 ) - ðåøåíèå (5.1).Ïîäñòàâèì åå â (5.1):nXi=1nn−1∂Φ X X ∂Φ ∂ΨjXiXi==∂xi∂Ψj ∂xii=1j=1=n−1X∂Φ∂Ψjj=1≡0nX|i=1∂Ψj∂xi{z }Xi∀Ψj â ñèëó Ò5.1Ñëåäîâàòåëüíî, ïî îïðåäåëåíèþ Φ - ðåøåíèå (5.1).2).

Ïîêàæåì, ÷òî ëþáîå ðåøåíèå óðàâíåíèÿ (5.1)z = Ψ(x1 , x2 , · · · , xn )ñîäåðæèòñÿ â îðìóëå (ìîæíî ïðåäñòàâèòü â âèäå)z = Φ (Ψ1 , · · · , Ψn−1 ) .Ïîäñòàâèì Ψ è Ψj (j = 1, 2, · · · , n − 1) â (5.1).Ïîëó÷èì ÑËÀÓ îòíîñèòåëüíî X1 , · · · , Xn : ∂Ψ∂ΨX1 ∂x1 + · · · + Xn ∂x≡ 0,n11≡ 0,+ · · · + Xn ∂ΨX1 ∂Ψ∂x1∂xn····················· ∂Ψn−1X1 ∂x1 + · · · + Xn ∂Ψ∂xn−1≡ 0.n55≡ 0 â D.Òàê êàênPi=1Xi2 6= 0 â D, òî îäíîðîäíàÿ ÑËÀÓ èìååò íåíóëåâîå ðåøåíèå. Îòñþäàñëåäóåò, ÷òî îïðåäåëèòåëü ∆ =Ψ, Ψ1 , · · · , Ψn−1 çàâèñèìû:D(Ψ,Ψ1 ,··· ,Ψn−1 )D(x1 ,x2 ,··· ,xn )= 0 â D, òî åñòü óíêöèèF (Ψ, Ψ1 , · · · , Ψn−1 ) = 0(*)Òàê êàê ïî óñëîâèþ 20 òåîðåìû ÿêîáèàíD (Ψ1 , · · · , Ψn−1 )6= 0,D(x1 , · · · , xn−1 )òî óðàâíåíèå (*) ìîæíî ðàçðåøèòü îòíîñèòåëüíî Ψ:z = Ψ = Φ (Ψ1 , · · · , Ψn−1 ) ,ãäå Φ - íåêîòîðàÿ äèåðåíöèðóåìàÿ óíêöèÿ, ò.å., äåéñòâèòåëüíî, óíêöèÿ z =Φ (Ψ1 , · · · , Ψn−1 ) ÿâëÿåòñÿ îáùèì ðåøåíèåì.5.