1-47 (Ответы на теоретический минимум)

Çàäàíèÿ ïî êóðñó ÄÓ. Òåîðìèí.217 ãðóïïà, Áîãîìîëîâ Àëåêñàíäð1. Ñôîðìóëèðóéòå òåîðåìó ñóùåñòâîâàíèÿ ðåøåíèÿ çàäà÷è Êîøè äëÿ óðàâíåíèÿïåðâîãî ïîðÿäêà.Th.1[Êîøè îa, |y − y0 | ≤ b}∃è ! ðåøåíèÿ]. Ïóñòü â íåêîòîðîì ïðÿìîóãîëüíèêåôóíêöèÿR = {|x − x0 | ≤f (x, y):1) íåïðåðûâíàÿ ïî ñîâîêóïíîñòè ïåðåìåííûõ;2) óäîâëåòâîðÿåò óñëîâèþ Ëèïøèöà, ò.å.:|f (x, y1 ) − f (x, y2 )| ≤ N |y1 − y2 |,y1 , y2 , y1 , y2 ∈ R.H = min(a, Mb ) (M = supR |f (x, y)|)ãäå N - ïîñòîÿííàÿ(íå çàâèñèò îò âûáîðàÒîãäà íà ñåãìåíòå|x − x0 | ≤ H,ãäåñóùåñòâóåò åäèí-ñòâåííîå ðåøåíèå çàäà÷è Êîøè:dy= f (x, y)dxy(x0 ) = y02.

Ñôîðìóëèðóéòå òåîðåìó ñóùåñòâîâàíèÿ ðåøåíèÿ çàäà÷è Êîøè äëÿ óðàâíåíèÿ y' =f(x,y). Ïðîâåðüòå å¼ âûïîëíåíèå äëÿ çàäà÷è.Th.1[Êîøè îÒîæå, ÷òî∃è ! ðåøåíèÿ]. Ïðîâåðêà âûïîëíåíèÿ:f (x, y) = 4x − 4y1) íåïðåðûâíîñòü f(x,y): î÷åâèäíà, íî âñå æå. Ïðîâåðèì ïî îïðåäåëåíèþ. Íóæíî äîêàçàòü,∀ε > 0∃δ > 0∀M, ρ(M (x, y), M0 (x0 , y0 )) < δ : |f (M ) − f (M0 )| < ε. Ôèêñèðóåì ε > 0.δ = 8ε . Òîãäà, íóæíî äîêàçàòü, ÷òî |f (x, y) − f (x0 , y0 )| < ε: |4x − 4y − 4x0 + 4y0 | =4|(x − x0 ) − (y − y0 )| ≤ 4(|x − x0 | + |y − y0 |) < 4(δ + δ) = 8δ < ε 2) óñëîâèå Ëèïøèöà÷òî:Ïîëîæèìïðîâåðÿåòñÿ ýëåìåíòàðíî:|f (x, y1 ) − f (x, y2 )| = 4|x − y1 − x + y2 | = 4|y1 − y2 | ≤ N |y1 − y2 |äëÿN ≥4Êàê âèäèì, âûáîð N íå çàâèñèò îò âûáîðà òî÷åê => óñëîâèå Ëèïøèöà âûïîëíÿåòñÿ.

Òàêèìîáðàçîì, óñëîâèÿ òåîðåìû âûïîëíåíû âñþäó äëÿ x>0.3. Ñôîðìóëèðóéòå òåîðåìó ñóùåñòâîâàíèÿ ðåøåíèÿ çàäà÷è Êîøè äëÿ óðàâíåíèÿ y' =f(x,y). Ïðîâåðüòå å¼ âûïîëíåíèå äëÿ çàäà÷è.Òîæå, ÷òîTh.1[Êîøè îf (x, y) =y√6∃è ! ðåøåíèÿ]. Ïðîâåðêà âûïîëíåíèÿ:1) íåïðåðûâíîñòü: î÷åâèäíà, íî çäåñü óæå ëó÷øå äîêàçûâàòü ÷åðåç ïðåäåëû.2) óñëîâèå Ëèïøèöà:√√| 6 y1 − 6 y2 | ≤ N |y1 − y2 |Ëåãêî âèäåòü, ÷òî âûáîð N çàâèñèò îòy1 , y2=> óñëîâèå Ëèïøèöà íå âûïîëíÿåòñÿ âñþäó,â òîì ÷èñëå è ïðè x>0.

Òåì íå ìåíåå ïîêàæåì ýòî:√√| 6 y1 − 6 y2 |≤N|y1 − y2 |ïðèy2 = 0èy1 → 0:√√| 6 y1 − 6 y2 |1= 5/6 → ∞|y1 − y2 |y114. Ñôîðìóëèðóéòå òåîðåìó ×àïëûãèíà ñóùåñòâîâàíèÿ è åäèíñòâåííîñòè ðåøåíèÿçàäà÷è Êîøè.α(x) è âåðõíåå β(x) ðåøåíèÿ çàäà÷è Êîα(x) < β(x). Ïóñòü ôóíêöèÿ f(x,y) íåïðåðûâíà è óäîâëåòâîðÿåò óñëîâèþïåðåìåííîé y , ò.å. ïðè êàæäîì x ∈ [0; a] :Th.2[×àïëûãèíà] Ïóñòü ñóùåñòâóåò íèæíååøè, òàêèå ÷òîËèïøèöà ïî|f (x, y1 ) − f (x, y2 )| ≤ N |y1 − y2 |, y1 , y2 ∈ [α(x), β(x)].Òîãäà çàäà÷à Êîøè èìååò åäèíñòâåííîå ðåøåíèå, ïðè÷åì:α(x) < y(x) < β(x), x ∈ [0; a]5. Äàéòå îïðåäåëåíèå ÔÑÐ ëèíåéíîãî îäíîðîäíîãî ÄÓ. Âèä îáùåãî ðåøåíèÿ.ÔÑÐ - ñîâîêóïíîñòü n ëèíåéíî íåçàâèñèìû íà îòðåçêå [a,b] ðåøåíèé ëèíåéíîãî îäíîðîäíîãî ÄÓ.Ñïðàâåäëèâà òåîðåìà.ThÏóñòüy1 (x), · · · , yn (x)- ÔÑÐ ëèíåéíîãî îäíîðîäíîãî óðàâíåíèÿ n-ãî ïîðÿäêà.Òîãäà ëþáîå ðåøåíèå ýòîãî óðàâíåíèå ìîæåò áûòü ïðåäñòàâëåíî â âèäå:z(x) =nXci yi (x),i=1ãäåci - ïîñòîÿííûå.Ïðèìåð:y 00 − 3y 0 = 0Ñîñòàâèì õàðàêòåðèñòè÷åñêîå óðàâíåíèå:λ2 − 3λ = 0Òîãäà ÔÑÐ óðàâíåíèÿ:y1 (x) = 1, y2 (x) = e3xÒîãäà, èç âûøå ñêàçàííîãî, îáùåå ðåøåíèå:z(x) = c1 + c2 e3x6.

Ìåòîä âàðèàöèè ïîñòîÿííîé äëÿ ðåøåíèÿ ÍË ÎÄÓ 1ï.Ñóòü ìåòîäà:Th.3[Ìåòîä âàðèàöèè ïîñòîÿííûõ] Ïóñòüy1 (x), .., yn (x)- ÔÑÐ îäíîðîäíîãî óðàâ-íåíèÿLy = y (n) + a1 (x)y (n−1) + ... + an (x)y = 0Py(x) = ni=1 ci (x)yi (x) áóäåò ðåøåíèåì íåîäíîðîäíîãîÒîãäà ôóíêöèÿc0 (x) óäîâëåòâîðÿåò ñëåäóþùåé ñèñòåìå:íèÿ, åñëè( P(j)nc0i (x)yi (x) = 0, äëÿ j = 0, 1, .., n − 2Pi=1(n−1)n0(x) = f (x)i=1 ci (x)yiÏðèìåð: x = tÎáùåå ðåøåíèå îäíîðîäíîãî:x(t) = at + b2ëèíåéíîãî óðàâíå-Âàðüèðóåì ïîñòîÿííûå:x(t)= a(t)t + b(t)Ïîëó÷èì:Òàêèì îáðàçîì,2a0 t + b 0 = 0a0 = t3b(t) = − t3 + C22a(t) = t2 + C13x(t) = ( t2 + C1 )t + (− t3 + C2 ),ãäå~ = (C1 , C2 )T Cïðîèçâîëüíûé âåêòîð.7.

Ìåòîä âàðèàöèè ïîñòîÿííîé äëÿ íåîäíîðîäíîé ëèíåéíîé íîðìàëüíîé ñèñòåìû.˙Íîðìàëüíàÿ ñèñòåìà - ñèñòåìà âèäà: ~x{f1 (t), .., fn (t)}= A(t)~x + F~(t),F~ (t) =~x(t) = {x1 (t), ..., xn (t)}.ãäå A(t) - ìàòðèöà nxn,- çàäàííàÿ âåêòîð ôóíêöèÿ(íåîäíîðîäíîñòü),Ñïðàâåäëèâà ñëåäóþùàÿ òåîðåìà:Th.4 Ïóñòü A(t) è X(t) íåïðåðûâíû íà [a,b] è èçâåñòíî ÔÑÐ îäíîðîäíî ëèíåéíîéíîðìàëüíîé ñèñòåìû. Òîãäà îáùåå ðåøåíèå íåîäíîðîäíîé ñèñòåìû íàõîäèòñÿ ñ ïîìîùüþêâàäðàòóð.˙~i (t) - îáùåå ðåøåíèåi=1 ci xPn ñîîòâåòñâóþùåé îäíîðîäíîé ñèñòåìû: ~x =xi (t) (7.1). Ïîäñòàâëÿåì â èñêîìîóþ ñèA(t)~x.

Âàðüèðóåì ïîñòîÿííûå => ~x = i=1 ci (t)~Ïóñòü~x =Pnñòåìó è ïîëó÷àåì:nXi=1c0i (t)~xi (t)+nX0ci (t)~xi (t) = A(t)|i=1P {zA(t)nXci (t)~xi (t) + F~(t)i=1}n~i (t)i=1 ci (t)xnXc0i (t)~xi (t) = F~i=1Åñëè ðàñïèñàòü ïî ñòðî÷íî, ïîëó÷èì ñèñòåìó: Pn 0ci (t)x1i = f1 (t) Pi=1n0i=1 ci (t)x2i = f2 (t)··· Pn 0i=1 ci (t)xni = fn (t)Òàê êàê îïðåäåëèòåëü ÂðîíñêîãîW [x~1 , · · · , x~n ] 6= 0=> ñèñòåìà ðàçðåøèìà èc0i (t) =φi (t), i = 1, n.Zci (t) =ãäåC̃ = (c˜1 , · · · , c˜n )Tφi (t)dt + c˜i ,- ïðîèçâîëüíûé âåêòîð.

Ïîäñòàâëÿÿ âûðàæåíèå äëÿ(7.1) è ïîëó÷àåì îáùåå ðåøåíèå äëÿ íåîäíîðîäíîé ËÍÑ.Ïðèìåð:ẋ = yẏ = −x +1cos(t)Îáùåå ðåøåíèå ñîîòâåòñâóþùåé îäíîðîäíîé: xc1 cos(t) + c2 sin(t)=y−c1 sin(t) + c2 cos(t)3ci (t)â ôîðìóëóÂàðüèðóåì ïîñòîÿííûå. Ïîëó÷èì: c1ln | cos(t)| + c˜1=c2t + c˜2Òàêèì îáðàçîì, îáùåå ðåøåíèå íåîäíîðîäíîé ñèñòåìû çàïèøåòñÿ â âèäå: x(ln | cos(t)| + c˜1 ) cos(t) + (t + c˜2 ) sin(t)=y−(ln | cos(t)| + c˜1 ) sin(t) + (t + c˜2 ) cos(t)8. Ïîêàæèòå ðàâíîñèëüíîñòü.Çàäà÷à Êîøè äëÿ ÎÄÓ n-ãî ïîðÿäêà ñòàâèòñÿ òàê:y (n) = f (x, y, y 0 , · · · , y (n−1) )ïðè íà÷àëüíûõ óñëîâèÿõ: y(x0 )0Ïóñòü òåïåðü y = y1 , y = y2 , · · ·ïðè íà÷àëüíûõ óñëîâèÿõ:0= y10 , y 0 (x0 ) = y20 , · · · , y (n−1) (x0 ) = yn−1., y (n−1) = yn .

Òîãäà ïðèõîäèì ê ñèñòåìå: 0 y10 = y2 y2 = y3···y 0 = yn n−1yn0 = f (x, y1 , y2 , · · · , yn )0.y1 (x0 ) = y10 , y2 (x0 ) = y20 , · · · , yn−1 (x0 ) = yn−1- óäîâëåòâîðÿþò âñåì óñëîâèÿì òåîðåìû î ∃ è !, ëèøü òîëüêî0(n)ïîòðåáóåì íåïðåðûâíîñòü f (x, y, y , · · · , y) è âûïîëíåíèå óñëîâèå Ëèïøèöà.Ïðàâûå ÷àñòè:fi = yi+1Ïðèìåð:mẍ = F (t, x, ẋ)Ïóñòüy1 = x, y2 = ẋ.9. Òåîðåìà∃ẏ1 = y2ẏ2 = F (t, y1 , y2 )è ! ðåøåíèÿ ç.Êîøè äëÿ ñèñòåìû ïåðâîãî ïîðÿäêà.Ïîñòàâèì çàäà÷ó Êîøè äëÿ ñèñòåìû óðàâíåíèé ïåðâîãî ïîðÿäêà (9.1).

Ïóñòü: ẋ1 = f1 (t, x1 , · · · , xn )···ẋn = fn (t, x1 , · · · , xn )Èëè â âåêòîðíîì âèäå:~x0 = A(t)~x.Ïóñòü çàäàíû íà÷àëüíûå óñëîâèÿ:xi (t0 ) = x0i (i = 1, n).Th [Êîøè] Ïóñòü:1)2)fifi- íåïðåðûâíû â îêðåñòíîñòè íà÷àëüíûõ çíà÷åíèé;óäîâëåòâîðÿþò óñëîâèþ Ëèïøèöà ïî âñåì àðãóìåíòàì, íà÷èíàÿ ñîÒîãäà ðåøåíèå çàäà÷è Êîøè (9.1)∃è !.10.Ôóíäàìåíòàëüíàÿ ìàòðèöà è òàê äàëåå.42-ãî.Ôóíäàìåíòàëüíàÿ ìàòðèöà - ìàòðèöà, ñîñòàâëåííàÿ èç ñòîëáöîâ, îáðàçóþùèõ ÔÑÐ -X(t) = (x~1 (t), · · · , x~n (t)).Òàê êàê êàæäûé ñòîëáåö ìàòðèöû - ðåøåíèå, òî ñïðàâåäëèâî ìàòðè÷íîå óðàâíåíèå (10.1):Ẋ(t) = A(t)X(t)Áóäåì èñêàòü ðåøåíèå â âèäå:~~x(t) = X(t)C,ãäå~C- ïðîèçâîëüíûé âåêòîð.Ïîäñòàâèâ â (10.1) => òîæäåñòâî => Ìû ïîñòðîèëè îáùåå ðåøåíèå îäíîðîäíîé ñèñòåìû.Ïðèìåð:ẋ = yẏ = −xËåãêî ïðîâåðèòü:x~1 =cos(t)− sin(t)èx~2 =sin(t)cos(t)îáðàçóþò ÔÑÐ ýòîé ñèñòåìû.

Òîãäà, îáùåå ðåøåíèå äàåòñÿ: xc1−c1 sin(t) + c2 cos(t)x~0 == X(T )C = (x~1 x~2 )=yc2c1 cos(t) + c2 sin(t)11. Òåîðåìà∃è ! ðåøåíèÿ ç.Êîøè äëÿ óðàâíåíèÿ n-ãî ïîðÿäêà.Ïîñòàâèì çàäà÷ó Êîøè (11.1).Ïóñòü:y (n) = f (x, y, y 0 , · · · , y (n−1) )Ïóñòü òàêæå íà÷àëüíûå óñëîâèÿ:0y(x0 ) = y10 , y 0 (x0 ) = y20 , · · · , y (n−1) (x0 ) = yn−1Òîãäà:Th [Êîøè] Ðåøåíèå çàäà÷è Êîøè (11.1) ∃ è !, åñëè â îêðåñòíîñòè íà÷àëüíûõ çíà÷åíèé(n−1)) ôóíêöèÿ f:(x0 , y0 , y00 , · · · , y01) ÿâëÿåòñÿ íåïðåðûâíîé ôóíêöèåé âñåõ ñâîèõ àðãóìåíòîâ;2) óäîâëåòâîðÿåò óñëîâèþ Ëèïøèöà ïî âñåì àðãóìåíòàì, íà÷èíàÿ ñî 2-ãî.12. Òåîðåìà î ñòðêòóðå ÔÑÐ îäíîðîäíîãî ëèíåéíîãî óðàâíåíèÿ n-ãî ïîðÿäêà ñïîñòîÿííûìè êîýôôèöèåíòàìè â ñëó÷àå ïðîñòûõ êîðíåé õàðàêòåðèñòè÷åñêîãîóðàâíåíèÿ.Ââåäåì ïàðó îïðåäåëåíèé.Ëèíåéíîå äèôôåðåíöèàëüíîå óðàâíåíèå n-ãî ïîðÿäêà - ÄÓ âèäà (11.1):Ly ≡ y (n) + a1 (x)y (n−1) + · · · + an (x)y = f (x)ïðè óñëîâèè, ÷òî âñåai (x)èf (x)- íåïðåðûâíû íà ìíîæåñòâå X(íåêîòîðîì ïîäìíîæåñòâå÷èñëîâîé ïðÿìîé).Îäíîðîäíîå óðàâíåíèå - óðàâíåíèå âèäà (11.1) ïðè ïðîòèâíîì ñëó÷àå ÄÓ íàçûâàåòñÿf (x) ≡ 0.íåîäíîðîäíûì.Ðåøåíèå óðàâíåíèÿ (11.1) - ëþáàÿ ôóíêöèÿy(x) ∈ C n [X],óäîâëåòâîðÿþùàÿ óðàâíå-íèþ.ÄÓ (11.1) íàçûâàåòñÿ óðàâíåíèåì ñ ïîñòîÿííûìè êîýôôèöèåíòàìè, åñëè5ai (x) = consti .Íåòðèâèàëüíûå ðåøåíèÿ ëèíåéíîãî îäíîðîäíîãî óðàâíåíèÿ ñ ïîñòîÿííûìè êîýôôèöèy(x) = Ceλx , ãäå Ñ6= 0, λ = const (ìåòîä Ýéëåðà).åíòàìè (ËÎÓñÏÊ) áóäåì ñòðîèòü â âèäå:Ïîäñòàâëÿÿ â ËÎÓñÏÊ:L[Ceλx ] = C [λn + a1 λn−1 + · · · + an ] eλx = 0{z}|M (λ)Îòñþäà:M (λ) = 0.Õàðàêòåðèñòè÷åñêèé ìíîãî÷ëåí - ìíîãî÷ëåíPnn−i.i=1 ai λÕàðàêòåðèñòè÷åñêðå óðàâíåíèå - óðàâíåíèåM (λ) = λn + a1 λn−1 + · · · + an = λ +M (λ) = 0.Ïóñòü: õàðàêòåðèñòè÷åñêîå óðàâíåíèå èìååò n ðàçëè÷íûõ (ïðîñòûõ) êîðíåéλxÎòñþäà, ëåãêî ïîêàçàòü, ÷òî êàæäîìó λi ñîîòâåòñâóåò ðåøåíèå: yi (x) = e i .λ1 , · · · , λn .Ñïðàâåäëèâà òåîðåìà:Th Ïóñòü êîðíè õàðàêòåðèñòè÷åñêîãî óðàâíåíèÿ ïðîñòûå.

Òîãäà ôóíêöèè:(iyi (x) = eλi x= 1, n)îáðàçóþò ÔÑÐ ýòîãî îäíîðîäíîãî ëèíåéíîãî óðàâíåíèÿ.Çàìå÷àíèå:(α+iβ)x (α−iβ)xÅñëè êîðíè λi - êîìïëåêñíûå, ïàðó ôóíêöèé e,e, îòâå÷àþùèõ λi =αxα − iβ , îáû÷íî çàìåíÿþò âåùåñòâåííûìè ôóíêöèÿìè e cos(βx), eαx sin(βx).α+iβ, λi+1 =Ïðèìåð:y 000 − 2y 00 − 3y = 0Õàðàêòåðèñòè÷åñêîå óðàâíåíèå (ÕÓ):λ3 − 2λ2 − 3λ = 0Åãî ðåøåíèÿ:λ1,2,3 = 0, 3, −1Òîãäà. ÔÑÐ îäíîðîäíîãî ëèíåéíîãî óðàâíåíèÿ:Y (x) = (1 e3x e−x )13. Ñôîðìóëèðóéòå îïðåäåëåíèå ìàòðèöû Êîøè îäíîðîäíîé ñèñòåìû ëèíåéíûõ ÎÄÓ.ÏóñòüX(t)- ôóíäàìåíòàëüíàÿ ìàòðèöà. Òîãäà:Ìàòðèöà Êîøè (ìàòðèöàíò) - ìàòðèöàK(t, τ ) = X(t)X −1 (τ ).Îíà îäíîçíà÷íî îïðåäå-ëÿåòñÿ êàê ðåøåíèå çàäà÷è Êîøè:dK(t, t0 )dt= A(t)K(t, t0 )K(t0 , t0 ) = EÑêàæåì ïàðó ñëîâ î ïîñòðîåíèè ìàòðèöàíòà è åãî èñïîëüçîâàíèè.Äëÿ ïîñòðîåíèÿ ìàòðèöû Êîøè íàäî ðåøèòü n âåêòîðíûõ çàäà÷ Êîøè: 0xi x~i = A(t)~x~ (t ) = x~0i i 0k = 1, nÐåøåíèå çàäà÷è Êîøè äëÿ íåîäíîðîäíîé ëèíåéíîé ñèñòåìû èìååò âèä:Zt~x(t) = K(t, t0 )x~0 +t0Ïðèìåð:ïîêà õç6K(t, τ )F~ (τ )dτ14.

Àëãîðèòì ðåøåíèÿ ëèíåéíîãî íåîäíîðîäíîãî ÎÄÓ n-ãî ïîðÿäêà ìåòîäîì Êîøè.Ââîäíûå îïðåäåëåíèÿ äàíû â 12.Ly = f (x)K(x, s) - ðåøåíèå ñïåöèàëüíîé çàäà÷è Êîøè: L[K(x, s)] = 0K(s, s) = K 0 (s, s) = · · · = K (n−2) (s, s) = 0 (n−1)K(s, s) = 1Ìåòîä Êîøè - ìåòîä íàõîæäåíèÿ ÷àñòíîãî ðåøåíèÿ:Ïóñòü ôóíêöèÿíàçûâàåòñÿ ôóíêöèåé Êîøè.Ñïðàâåäëèâà ñëåäóþùàÿ òåîðåìà:RxK(x, s)f (s)ds - ðåøåíèå çàäà÷è Êîøè äëÿ íåîäíîðîäíîãî óðàâTh Ôóíêöèÿ y(x) =x0íåíèÿ ñ íóëåâûìè íà÷àëüíûìè óñëîâèÿìè, ò.å: Ly = f (x)y(x0 ) = y 0 (x0 ) = · · · = y (n−1) (x0 ) = 0x, x0 ∈ [a, b]ÔóíêöèÿK(x, s)- ôóíêöèÿ Êîøè.ProofÏðîâåðÿåì íåïîñðåäñòâåííî ïîäñòàíîâêîé y(x) â óðàâíåíèå. Ïðåæäå:Zxan (x)|y =K(x, s)f (s)dsx0Z0an−1 (x)|y = K(x, x) f (x) +| {z }=0xKx0 (x, s)f (s)dsx0··· Zxa1 (x)|y (n−1) = Kx(n−2) f (x) +Kx(n−1) (x, s)f (s)ds| {z }x0=0Z x(n)(n−1)f (x) +|y = KxKx(n) (x, s)f (s)ds| {z }x0=1Óìíîæèì êàæäîåy(k)(x)íàak (x),ñëîæèì è ïîëó÷èì:ZxLy = f (x) +L[K(x, s)]f (s)dx = f (x)x0Îòñþäà ñðàçó: ÷òä.Ïðèìåð: 00 y + y = f (x)y(0) = 0 0y (0) = 0Ðåøàåì îäíîðîäíîå:y(x) = c1 sin(t) + c2 cos(t)Òåïåðü èç ñâîéñòâ ôóíêöèè Êîøè:c1 sin(s) + c2 cos(s) = 0c1 cos(s) − c2 sin(s) = 1Îòñþäà ñðàçó: c1cos(s)=c2− sin(s)Òî åñòü:K(x, s) = sin(x − s).Òîãäà ðåøåíèå äà¼òñÿ:7y(x) =Rx0sin(x − s)f (s)ds.15.

Îïðåäåëåíèå è ñâîéñòâà ôóíäàìåíòàëüíîé ìàòðèöû îäíîðîäíîé ëèíåéíîé ñèñòåìûÎÄÓ.Ïîêà õç, ÷åì îòëè÷àåòñÿ îí îòñì. âîïðîñ 10.16. Îïðåäåëåíèå è ñâîéñòâà îïðåäåëèòåëÿ Âðîíñêîãî, ïîñòðîåííîãî èç ðåøåíèéîäíîðîäíîãî ÎÄÓ n-ãî ïîðÿäêà.Îïðåäåëèòåëü Âðîíñêîãî (Âðîíñêèàí) ôóíêöèéy1 , y2 , · · · , yn((n-1) ðàç äèôôåðåí-öèðóåìûõ íà ïðîìåæóòêå X) - îïðåäåëèòåëü âèäà:W (x) = W [y1 , y2 , · · · , yn ] = Ïóñòü òåïåðüy1 , y2 , · · · , yny1y10···(n−1)y1y2y20···(n−1)y2············ynyn0···(n−1)yn- ðåøåíèÿ îäíîðîäíîãî ÎÄÓ n-ãî ïîðÿäêà. Òîãäà ñïðàâåäëèâûñëåäóþùèå òåîðåìû-ñâîéñòâà:y1 , y2 , · · · , yn - ëèíåéíîW [y1 , · · · , yn ] ≡ 0 íà ýòîì îòðåçêå.1) Åñëè ôóíêöèèìûçàâèñèìû ïðèx ∈ [a; b],òî âðîíñêèàí ýòîé ñèñòå-(äîêàçûâàåòñÿ "â ëîá"âûïèñûâàíèåì óñëîâèÿ ëèíåéíîé çàâèñèìîñòè âåêòîðîâ è äèôôåðåíöèðîâàíèåì îíîãî (n-1) ðàç)2) Åñëè ëèíåéíî íåçàâèñèìûå (ëíç) ôóíêöèèy1 , · · · , ynÿâëÿþòñÿ ðåøåíèÿìè ëèíåéíîãîîäíîðîäíîãî óðàâíåíèÿ:y (n) + p1 (x)y (n−1) + · · · + pn (x)y = 0ñ íåïðåðûâíûìè íà îòðåçêå [a;b] êîýôôèöèåíòàìèpi (x),òî âðîíñêèàí ýòîé ñèñòåìûW [y1 , · · · , yn ] 6= 0íè â îäíîé òî÷êå ýòîãî îòðåçêà.Ïðèìåð:y 00 − y = 0ÕÓ:λ2 = 1ÔÑÐ:Ñîñòàâèì âðîíñêèàí xy1e= −xy2eW [y1 , y2 ]: x −x e eW [y1 , y2 ] = xe −e−x = −1 − 1 = −2Êàê è îæèäàëîñü, âðîíñêèàí ëíç âåêòîðîâ îòëè÷åí îò íóëÿ âñþäó, ãäå âåêòîðà ëíç.17.