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Ëèíåéíûå óðàâíåíèÿ ïåðâîãî ïîðÿäêàÐàññìîòðèì óðàâíåíèå âèäày 0 = p(x)y + q(x).Ïîëàãàåì, ÷òî ôóíêöèè p è q íåïðåðûâíû íà èíòåðâàëå (a, b) (p, q ∈ C((a, b))), òîãäàïðàâàÿ ÷àñòü ä.ó. f (x, y) = p(x)y +q(x) íåïðåðûâíà â ïîëîñå D = (a, b)×R, å¼ ÷àñòíàÿ= p(x) ∈ C(D),ïðîèçâîäíàÿ ïî ïåðåìåííîé y òàêæå íåïðåðûâíà â ïîëîñå D: ∂f∂yò.å. âûïîëíÿþòñÿ óñëîâèÿ òåîðåìû Ïèêàðà. Çíà÷èò, ìîæåì ñðàçó ñêàçàòü, ÷òî ÷åðåçêàæäóþ òî÷êó ïîëîñû D ïðîõîäèò ãðàôèê íåïðîäîëæàåìîãî ðåøåíèÿ, è òîëüêî îäèí.Îïðåäåëåíèå.
Óðàâíåíèåy 0 = p(x)y(2)ëèíåéíûì îäíîðîäíûì óðàâíåíèåì,íàçûâàåòñÿóðàâíåíèåy 0 = p(x)y + q(x)ëèíåéíûì íåîäíîðîäíûì óðàâíåíèåì(3)íàçûâàåòñÿ.Ñëåäóåò îòìåòèòü, ÷òî ñëîâî "îäíîðîäíîå" â ïðîøëîì ïàðàãðàôå è â ýòîì îçíà÷àåò ðàçíûå óðàâíåíèÿ. Óðàâíåíèÿ, ðàññìàòðèâàåìûå çäåñü, õàðàêòåðèçóþòñÿ ñðàçóïàðîé ñëîâ "ëèíåéíîå îäíîðîäíîå" ëèáî "ëèíåéíîå íåîäíîðîäíîå". Ëèíåéíîñòü îçíà÷àåò, ÷òî åñëè âçÿòü äâà ðåøåíèÿ óðàâíåíèé ñ ðàçíîé íåîäíîðîäíîñòüþ (ò.å. ôóíêöèåé q ), òî ëèíåéíàÿ èõ êîìáèíàöèÿ åñòü ðåøåíèå óðàâíåíèÿ ñ ëèíåéíîé êîìáèíàöèåéíåîäíîðîäíîñòåé. Áîëåå òî÷íî ýòî ñâîéñòâî ðåøåíèé çàïèñûâàåòñÿ â âèäå ëåììû.Ïóñòü ðåøåíèå íåîäíîðîäíîãî óðàâíåíèÿ y0 = p(x)y + q1(x), ðåøåíèå íåîäíîðîäíîãî óðàâíåíèÿ y0 = p(x)y + q2(x),òîãäà äëÿ ïðîèçâîëüíûõ êîíñòàíò C1 è C2 ôóíêöèÿy(x) = C1 y1 (x)+C2 y2 (x) ÿâëÿåòñÿ ðåøåíèåì óðàâíåíèÿ y 0 = p(x)y +C1 q1 (x)+C2 q2 (x).Ëåììà (ïðèíöèï ñóïåðïîçèöèè).y1 (x)y2 (x)Äîêàçàòåëüñòâî.Ïîñêîëüêó y1 (x) è y2 (x) ðåøåíèÿ ñîîòâåòñòâóþùèõ óðàâíåíèé, òî âûïîëíÿþòñÿòîæäåñòâà y10 (x) ≡ p(x)y1 (x) + q1 (x) è y20 (x) ≡ p(x)y2 (x) + q2 (x).
Íàì íàäî ïîêàçàòüâûïîëíåíèÿ òîæäåñòâà y 0 (x) ≡ p(x)y(x) + C1 q1 (x) + C2 q2 (x), ÷òî è áóäåò îçíà÷àòü, ÷òîôóíêöèÿ y(x) ÿâëÿåòñÿ ðåøåíèåì íóæíîãî óðàâíåíèÿ. Èìååìy 0 (x) ≡ C1 y10 (x) + C2 y20 (x) ≡ C1 (p(x)y1 (x) + q1 (x)) + C2 (p(x)y2 (x) + q2 (x)) ≡≡ p(x)(C1 y1 (x) + C2 y2 (x)) + C1 q1 (x) + C2 q2 (x) ≡ p(x)y(x) + C1 q1 (x) + C2 q2 (x).Ïîëó÷èëè íóæíîå òîæäåñòâî.×ÒÄÐàññìîòðèì òåïåðü îäíîðîäíîå ëèíåéíîå óðàâíåíèå y 0 = p(x)y . Óòâåðæäåíèå ñëåäóþùåé òåîðåìû ñðàçó ñëåäóåò èç îáùåãî ðåøåíèÿ äëÿ ýòîãî ä.ó. (à ïîñêîëüêó ýòîóðàâíåíèå ñ ðàçäåëÿþùèìèñÿ ïåðåìåííûìè, ðåøàåòñÿ îíî ïðîñòî), îäíàêî äëÿ ïðèâûêàíèÿ è ïðèìåíåíèÿ ê óðàâíåíèÿì âûñîêîãî ïîðÿäêà ìû å¼ ðàññìîòðèì.11Òåîðåìà 1 (îá îáùåì ðåøåíèè îäíîðîäíîãî óðàâíåíèÿ).
Ìíîæåñòâî ðåøåíèé ëèíåéíîãî îäíîðîäíîãî óðàâíåíèÿ ëèíåéíîå ïðîñòðàíñòâî ðàçìåðíîñòè 1.Äîêàçàòåëüñòâî.Âî-ïåðâûõ, íàäî ïîêàçàòü ëèíåéíîñòü ïðîñòðàíñòâà ðåøåíèé, ò.å. ïîêàçàòü, ÷òîåñëè y1 (x) è y2 (x) ïðîèçâîëüíûå ðåøåíèÿ óðàâíåíèÿ y 0 = p(x)y , òî èõ ëèíåéíàÿ êîìáèíàöèÿ y(x) = C1 y1 (x) + C2 y2 (x) ÿâëÿåòñÿ ðåøåíèåì òîãî æå ñàìîãî óðàâíåíèÿ. Ýòîñðàçó æå ñëåäóåò èç ïðèíöèïà ñóïåðïîçèöèè.Âî-âòîðûõ, íàäî ïîêàçàòü, ÷òî ïðîñòðàíñòâî ðåøåíèé îäíîìåðíî, ò.å.
åñëè âîçüì¼ì êàêîå-íèáóäü íåíóëåâîå ðåøåíèå ye(x) óðàâíåíèÿ y 0 = p(x)y , òî ëþáîå äðóãîåðåøåíèå y(x) ýòîãî óðàâíåíèÿ ïðåäñòàâèìî â âèäå y(x) = C ye(x), ãäå C íåêîòîðàÿ:êîíñòàíòà. Äëÿ äîêàçàòåëüñòâà ðàññìîòðèì ïðîèçâîäíóþ îòíîøåíèÿ y(x)ye(x)y(x)ye(x)0≡y 0 (x)ey (x) − y(x)ey 0 (x)p(x)y(x)ey (x) − y(x)p(x)ey (x)≡≡ 0,22ye (x)ye (x)çíà÷èò, y(x) = C ye(x).×ÒÄÑëåäóþùàÿ òåîðåìà äà¼ò îòâåò íà âîïðîñ, êàê âûãëÿäÿò ðåøåíèÿ íåîäíîðîäíîãîóðàâíåíèÿ.Òåîðåìà 2 (î ðåøåíèè íåîäíîðîäíîãî óðàâíåíèÿ).y0y = p(x)y + q(x)y0y = p(x)yyy 0 = p(x)y + q(x)y =y +yÏðèìåð. Óðàâíåíèå y 0 = y+x ÿâëÿåòñÿ ëèíåéíûì íåîäíîðîäíûì, ìû óæå ðåøàëèåãî â ïðîøëîì ïàðàãðàôå.
Åãî îáùåå ðåøåíèå èìååò âèä yîí = Cex − x − 1 = yîî +y÷àñò . Çäåñü yîî = Cex îáùåå ðåøåíèå y 0 = y , y÷àñò = −x − 1 îäíî èç ðåøåíèéíåîäíîðîäíîãî y 0 = y + x.Äîêàçàòåëüñòâî òåîðåìû.Âî-ïåðâûõ, ñóììà yîî + y÷àñò ÿâëÿåòñÿ ðåøåíèåì íåîäíîðîäíîãî óðàâíåíèÿ èçïðèíöèïà ñóïåðïîçèöèè: yîî ðåøåíèå y 0 = p(x)y + 0, y÷àñò ðåøåíèå y 0 = p(x)y +q(x), à ëèíåéíàÿ êîìáèíàöèÿ 1 · yîî + 1 · y÷àñò ÿâëÿåòñÿ ðåøåíèåì y 0 = p(x)y + 1 · 0 +1 · q(x).Âî-âòîðûõ, ïîêàæåì, ÷òî ïðîèçâîëüíîå ðåøåíèå y(x) íåîäíîðîäíîãî óðàâíåíèÿ0y = p(x)y + q(x) ïðåäñòàâèìî â âèäå íóæíîé ñóììû.
Ïîñêîëüêó y(x) ðåøåíèåy 0 = p(x)y + q(x), y÷àñò ðåøåíèå y 0 = p(x)y + q(x), òî ðàçíîñòü y(x) − y÷àñò (x) ðåøåíèå y 0 = p(x)y+q(x)−q(x), ò.å. ëèíåéíîãî îäíîðîäíîãî óðàâíåíèÿ, è èç òåîðåìû 1eye(x) äëÿ íåêîòîðîé êîíñòàíòû Ce.  ðåçóëüòàòå, y(x) =ñëåäóåò, ÷òî y(x)−y÷àñò (x) = Ceye(x) + y÷àñò (x), ò.å. ïðîèçâîëüíîå ðåøåíèå íåîäíîðîäíîãî îáÿçàòåëüíî ïðåäñòàâèìîCâ âèäå ñóììû.×ÒÄíåîäíîðîäíîãî óðàâíåíèÿðîäíîãî óðàâíåíèÿ, ò.å. îíÎáùåå ðåøåíèå îíåñòü ñóììà îáùåãî ðåøåíèÿ îî îäíîè ÷àñòíîãî ðåøåíèÿ ÷àñò íåîäíîðîäíîãî óðàâíåíèÿîî ÷àñò.Àëãîðèòì ðåøåíèÿ ëèíåéíîãî íåîäíîðîäíîãî óðàâíåíèÿ y 0 = p(x)y + q(x)I.
Èùåì îáùåå ðåøåíèå îäíîðîäíîãî óðàâíåíèÿ y 0 = p(x)y .Ýòî óðàâíåíèå ñ ðàçäåëÿþùèìèñÿ ïåðåìåííûìè.1. y(x) ≡ 0 ðåøåíèå.122. Ïóñòü y 6= 0, òîãäày 0 (x)= p(x),y(x)Zx 0Zxy (ξ)dξ = p(ξ)dξ,y(ξ)x0x0Zy(x)dη=ηZxp(ξ)dξ,x0y(x0 )Zxln |y(x)| − ln |y(x0 )| =p(ξ)dξ.x0 íà÷àëå ïàðàãðàôà ïîêàçûâàëîñü, ÷òî ãðàôèêè ðåøåíèé çàìîùàþò âñþ ïîëîñóD = (a, b) × R, çíà÷èò, x è x0 ïðèíàäëåæàò èíòåðâàëó (a, b), à y(x0 ) ìîæíî áðàòüèç èíòåðâàëîâ (−∞, 0) è (0, +∞). Äàëåå,Rx|y(x)| = |y(x0 )|ex0Rxp(ξ)dξ,p(ξ)dξy(x) = ±|y(x0 )|ex0Rxy(x) = C0 ex0p(ξ)dξ,, ãäå C0 = ±|y(x0 )| ∈ R \ {0}.Îáúåäèíÿÿ ðåçóëüòàòû ïóíêòîâ 1 è 2, ïîëó÷àåì îáùåå ðåøåíèå îäíîðîäíîãî óðàâíåíèÿ:xRp(ξ)dξyîî (x) = Cex0, ãäåC ∈ R.Íåíóëåâîå ðåøåíèå ye îäíîðîäíîãî óðàâíåíèÿ èç òåîðåìû 1 ìîæíî âçÿòü ñëåäóþùèìRxp(ξ)dξye(x) = e, òîãäà yîî (x) = C ye(x), ãäå C ∈ R.II.
Èùåì ÷àñòíîå ðåøåíèå íåîäíîðîäíîãî óðàâíåíèÿ y 0 = p(x)y + q(x).x0Ìåòîä âàðèàöèè ïðîèçâîëüíîé ïîñòîÿííîé.×àñòíîå ðåøåíèå èùåì ïî÷òè â âèäå îáùåãî ðåøåíèÿ îäíîðîäíîãî óðàâíåíèÿ,òîëüêî âìåñòî êîíñòàíòû C áåð¼ì ôóíêöèþ, ò.å. y÷àñò (x) = u(x)ey (x), è òåïåðü íàäîíàéòè ïîäõîäÿùóþ u(x), ÷òîáû ïðè ïîäñòàíîâêå ýòîãî ïðîèçâåäåíèÿ â íåîäíîðîäíîåóðàâíåíèå ïîëó÷àëîñü òîæäåñòâî. ëåâîé ÷àñòè íåîäíîðîäíîãî óðàâíåíèÿ:0y÷àñò(x) = u0 (x)ey (x) + u(x)ey 0 (x) = u0 (x)ey (x) + u(x)p(x)ey (x),â ïðàâîé ÷àñòè íåîäíîðîäíîãî óðàâíåíèÿ:p(x)y÷àñò (x) + q(x) = p(x)u(x)ey (x) + q(x),13òîæäåñòâî ïîëó÷èòñÿ, åñëèu0 (x)ey (x) = q(x),îòñþäà íàõîäèòñÿ u(x):q(x),ye(x)ZxZxq(ξ)u0 (ξ)dξ =dξ,ye(ξ)u0 (x) =x0x0Zxu(x) =x0Òàêèì îáðàçîì, ôóíêöèÿRxx0q(ξ)dξ + u(x0 ).ye(ξ)!q(ξ)dξye(ξ)+ u0 ye(x) ÿâëÿåòñÿ ðåøåíèåì ëèíåéíîãî íåîäíî-ðîäíîãî óðàâíåíèÿ ïðè ïðîèçâîëüíîé êîíñòàíòå u0 .III.
Èùåì îáùåå ðåøåíèå íåîäíîðîäíîãî óðàâíåíèÿ y 0 = p(x)y + q(x).Ïðèìåíÿÿ óòâåðæäåíèå òåîðåìû 2 è ðåçóëüòàòû äâóõ ïðåäûäóùèõ ïóíêòîâ, îáùååðåøåíèå íåîäíîðîäíîãî óðàâíåíèÿ çàïèñûâàåòñÿ â âèäå xZq(ξ)dξ + u0 ye(x)yîí (x) = C ye(x) + ye(ξ)x0Ïåðâîå è òðåòüå ñëàãàåìûå ìîæíî îáúåäèíèòü, ïîñêîëüêó C è u0 ïðîèçâîëüíûå êîíñòàíòû.
 èòîãåZxq(ξ)dξey (x),yîí (x) = C ye(x) +ye(ξ)x0 òàêîé çàïèñè ïîíÿòíî, ÷òî ïðè íàõîæäåíèè ÷àñòíîãî ðåøåíèÿ y÷àñò ìîæíî áûëîêîíñòàíòó u0 âçÿòü ëþáûì êîíêðåòíûì ÷èñëîì, â ÷àñòíîñòè íóë¼ì. Îáùåå ðåøåíèåíàéäåíî.Ïðèìåð. Ðåøèì ëèíåéíîå íåîäíîðîäíîå óðàâíåíèåy0 =2y+ x2 .xÑîãëàñíî àëãîðèòìó íàõîäèì îáùåå ðåøåíèå îäíîðîäíîãî óðàâíåíèÿy0 =2y.xÝòî óðàâíåíèå ñ ðàçäåëÿþùèìèñÿ ïåðåìåííûìè, ïîýòîìó ñðàçó íàõîäèì ðåøåíèå0y(x) = 0, ïîñëå ÷åãî ñ÷èòàåì, ÷òî y 6= 0, ðàçäåëÿåì ïåðåìåííûå yy = x2 è èíòåãðèðóåìZZdy2=dx,yx14y(x) = Cx2 , C ∈ R \ {0}.Ó÷èòûâàÿ ðåøåíèå y(x) = 0, ïîëó÷àåìyîî (x) = Cx2 , C ∈ R.Òåïåðü èùåì ÷àñòíîå ðåøåíèå óðàâíåíèÿ y 0 = 2y+x2 ìåòîäîì âàðèàöèè ïðîèçâîëüíîéxïîñòîÿííîé, ò.å.â âèäåy÷àñò (x) = u(x)x2 .Ïîäñòàâèì ýòî ïðåäñòàâëåíèå â óðàâíåíèå:0(x) = u0 (x)x2 + u(x)2x,y÷àñòïîëó÷àåì2y÷àñò2u(x)x2+ x2 =+ x2 ,xx2u(x)x2+ x2 ,xu0 (x)x2 = x2 ,u0 (x)x2 + u(x)2x =u0 (x) = 1,u(x) = x + u0 .Òàêèì îáðàçîì, ïðîèçâîëüíîå ÷àñòíîå ðåøåíèå èìååò âèäy÷àñò (x) = (x + u0 )x2 ,à ïîñêîëüêó íàì íàäî îäíî ÷àñòíîå ðåøåíèå, òî ìîæåì âçÿòü u0 , íàïðèìåð, ðàâíîåíóëþ, òîãäày÷àñò (x) = x3 . èòîãå âñå ðåøåíèÿ íåîäíîðîäíîãî óðàâíåíèÿ y 0 =îáðàçîì:y(x) = Cx2 + x3 .2yx+x2 çàïèñûâàþòñÿ ñëåäóþùèì6.
Óðàâíåíèå Áåðíóëëè. Óðàâíåíèå Ðèêêàòè.Óðàâíåíèå âèäày 0 = p(x)y + q(x)y α ,óðàâíåíèåì Áåðíóëëèãäå α 6= 0, 1, íàçûâàåòñÿ. Åñëè α = 0, òî ýòî ëèíåéíîå íåîäíîðîäíîå óðàâíåíèå, åãî íàó÷èëèñü ðåøàòü â ïðîøëîì ïàðàãðàôå, åñëè α = 1, òî ýòîóðàâíåíèå ñ ðàçäåëÿþùèìèñÿ ïåðåìåííûìè.Ïîëàãàåì, ÷òî ôóíêöèè p è q íåïðåðûâíû íà èíòåðâàëå (a, b), ò.å. p, q ∈ C((a, b)).Ïðè α > 0 ïðàâàÿ ÷àñòü ä.ó. îïðåäåëåíà â ïîëîñå (a, b) × R, y(x) = 0 ðåøåíèå. Ïðèα < 0 ïðàâàÿ ÷àñòü ä.ó. îïðåäåëåíà íà (a, b) × R \ {y = 0}.15Äëÿ ðåøåíèÿ óðàâíåíèÿ Áåðíóëëè ñäåëàåì çàìåíó, êîòîðàÿ ïðèâåä¼ò ê ëèíåéíîìóóðàâíåíèþ.
Ñíà÷àëà ðàçäåëèì åãî íà y α (çàïîìíèâ ïðè ýòîì, ÿâëÿåòñÿ ëè y(x) = 0ðåøåíèåì), ïîëó÷èìy01= p(x) α−1 + q(x).αyyÌîæíî çàìåòèòü, ÷òî0y 0 (x)11=yα1 − α y(x)α−1òîãäà óðàâíåíèå ïðåîáðàçóåòñÿ â0111= p(x) α−1 + q(x).α−11 − α y(x)y1Çäåñü ìîæíî çàìåòèòü, êàêóþ çàìåíó íàäî ïðîâåñòè. Ýòî z(x) = α−1 , òîãäà óðàây (x)íåíèå ñòàíåò1z 0 = p(x)z + q(x)1−αèëèz 0 = (1 − α)p(x)z + (1 − α)q(x),ò.å. ëèíåéíûì íåîäíîðîäíûì.Óðàâíåíèå âèäày 0 = p(x)y + q(x)y 2 + r(x)óðàâíåíèåì Ðèêêàòèíàçûâàåòñÿ. Òàêîå óðàâíåíèå óæå ñëîæíåå, åãî ðåøåíèÿ íåâñåãäà âîçìîæíî ïðåäñòàâèòü â âèäå ýëåìåíòàðíûõ ôóíêöèé, îäíàêî ìîæíî çàìåòèòü, ÷òî åñëè íàéäåíî êàêîå-ëèáî åãî ÷àñòíîå ðåøåíèå y÷àñò (x), òî çàìåíà w(x) =y(x) − y÷àñò ñâåä¼ò ýòî óðàâíåíèå ó óðàâíåíèþ Áåðíóëëè.
Äåéñòâèòåëüíî, ïîäñòàâèìy(x) = w(x) + y÷àñò â óðàâíåíèå Ðèêêàòè:(w + y÷àñò )0 = p(x)(w + y÷àñò ) + q(x)(w + y÷àñò )2 + r(x)02w0 + y÷àñò= p(x)w + p(x)y÷àñò + q(x)w2 + 2q(x)wy÷àñò + q(x)y÷àñò+ r(x).Ïîñêîëüêó y÷àñò ÿâëÿåòñÿ ÷àñòíûì ðåøåíèåì, òî âûïîëíÿåòñÿ òîæäåñòâî02y÷àñò≡ p(x)y÷àñò + q(x)y÷àñò+ r(x),è óðàâíåíèå Ðèêêàòè ïåðåéä¼ò âw0 = p(x)w + q(x)w2 + 2q(x)wy÷àñò .À ýòî óðàâíåíèå Áåðíóëëè:w0 = (p(x) + 2q(x)y÷àñò )w + q(x)w2 .Ïðèìåð. Ðåøèì óðàâíåíèå Ðèêêàòèy 0 = −xy + y 2 + 1.16Ìîæíî çàìåòèòü, ÷òî y(x) = x ðåøåíèå ýòîãî ä.ó., ïîýòîìó ñäåëàåì çàìåíó w(x) =y(x) − x. Ïîäñòàâèì y(x) = w(x) + 1 â óðàâíåíèå:w0 + 1 = −x(w + x) + (w + x)2 + 1.Ðàñêðîåì ñêîáêè, ïðèâåä¼ì îáùèå ñëàãàåìûå è ïîëó÷èìw0 = xw + w2 .Ýòî óðàâíåíèå Áåðíóëëè. Íàäî äåëàòü åù¼ çàìåíó.
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