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Òàêèì îáðàçîì, ôóíêöèÿ z(x), îïðåäåë¼ííàÿ íàhα, ω+eh], ÿâëÿåòñÿ ðåøåíèåì çàäà÷è Êîøè(x0 , y0 ), â òî âðåìÿ êàê ôóíêöèÿ y(x), îïðåäåë¼ííàÿ íà hα, ω], ÿâëÿåòñÿ íåïðîäîëæàåìûì ðåøåíèåì çàäà÷è Êîøè(x0 , y0 ). Ïîëó÷èëè ïðîòèâîðå÷èå. Çíà÷èò, ω ∈/ hα, ωi. Àíàëîãè÷íî äîêàçûâàåòñÿ, ÷òî α ∈/ hα, ωi.Ëåììà 4 äîêàçàíà.32Òåîðåìà Ïèêàðà äîêàçàíà.11. Ïîâåäåíèå íåïðîäîëæàåìûõ ðåøåíèéÍàïîìíèì, ÷òî êîìïàêò ýòî îãðàíè÷åííîå çàìêíóòîå ìíîæåñòâî.Ïóñòüf ∈ C(D),∈ C(D), D íåïóñòîå îòêðûòîå ìíîæåñòâî, òî÷êà (x0 , y0 ) ∈ D,ôóíêöèÿ y(x) íåïðîäîëæàåìîå ðåøåíèå çàäà÷è Êîøè(x0, y0), îïðåäåë¼ííîå íà (α, ω),Γ ïðîèçâîëüíûé êîìïàêò â D, ñîäåðæàùèé (x0 , y0 ),òîãäàñóùåñòâóþò ÷èñëà r−, r+ ∈ R òàêèå, ÷òî (x, y(x)) ∈/ Γ ïðè x ∈ (α, r−) ∪ (r+, ω)(èíûìè ñëîâàìè, ãðàôèê íåïðîäîëæàåìîãî ðåøåíèÿ âñåãäà âûéäåò çà ãðàíèöû ïðîèçâîëüíîãî êîìïàêòà).Òåîðåìà î ïîêèäàíèè êîìïàêòà.∂f∂y{íóæåí ðèñóíîê}Êàê ñëåäñòâèå ìîæíî çàêëþ÷èòü, ÷òî òî÷êè (α, y(α)) è (ω, y(ω)) íå ïðèíàäëåæàòìíîæåñòâó D.Ñ èñïîëüçîâàíèåì ýòîé òåîðåìû äîêàçûâàåòñÿ ñëåäóþùàÿÏóñòü, ∂f∂y ∈ C(D), D = (a, b) × R âåðòèêàëüíàÿ ïîëîñà,ôóíêöèÿ y(x) íåïðîäîëæàåìîå ðåøåíèå çàäà÷è Êîøè(x0, y0), îïðåäåë¼ííîå íà (α, ω),òîãäàI) ëèáî a = α, ëèáî a < α, è òîãäà |y(x)| → +∞ ïðè x → α + 0;II) ëèáî b = ω, ëèáî b > ω, è òîãäà |y(x)| → +∞ ïðè x → ω − 0.Òåîðåìà î ïîâåäåíèè â ïîëîñå.f ∈ C(D)Äîêàæåì II.Íàäî ïîêàçàòü, ÷òî åñëè b > ω , òîãäà |y(x)| → +∞ ïðè x → ω − 0.
Èñïîëüçóåìäëÿêîìïàêòà.Ðàññìîòðèì ñåìåéñòâî êîìïàêòîâ ΓR = ýòîãî òåîðåìó î ïîêèäàíèè,|y|6R.Ïîòåîðåìåî ïîêèäàíèè êîìïàêòà ñóùåñòâóåò(x, y) ∈ R : x ∈ x0 , ω+d2, çíà÷èò,r− (R) ∈ R, ïðè êîòîðîì (x, y(x)) ∈/ Γ ïðè x ∈ (r+ (R), ω). Ïîñêîëüêó ω < ω+d2|y(r+ (R))| > R, |y(x)| > R ïðè x ∈ (r+ (R), ω). Óâåëè÷èâàÿ R, ïîëó÷àåì |y(x)| → +∞ïðè x → ω − 0. Ïóíêò I äîêàçûâàåòñÿ àíàëîãè÷íî.×ÒÄÇàìå÷àíèå ê òåîðåìå î ïîâåäåíèè â ïîëîñå. Âûðàæåíèå |y(x)| → +∞ ïðèx → ω − 0 îçíà÷àåòëèáî y(x) → +∞ ïðè x → ω − 0,ëèáî y(x) → −∞ ïðè x → ω − 0.
Òàê, åñëè äëÿ íåêîòîðîãî R > 0 y(r+ (R)) > R, òî ïîòåîðåìå î ïîêèäàíèè êîìïàêòà |y(x)| > R ïðè x ∈ (r+ (R), ω). Åñëè íàéä¼òñÿ x1 ïðèêîòîðîì y(x1 ) < −R, òî èç íåïðåðûâíîñòè ôóíêöèè y íàéä¼òñÿ òî÷êà x2 , äëÿ êîòîðîéy(x2 ) = 0. Ïîëó÷èëè ïðîòèâîðå÷èå. Çíà÷èò, y(x) > R ïðè âñåõ x ∈ (r+ (R), ω).12. Ìåòîäû ïîíèæåíèÿ ïîðÿäêàÐàññìîòðèì óðàâíåíèå n-ãî ïîðÿäêàG x, y, y 0 , . . . , y (n) = 0,(5)ãäå y = y(x), G : D → R, D ⊂ R(n+2) .
Ðàññìîòðèì íåñêîëüêî ìåòîäîâ, ïîçâîëÿþùèõïîíèçèòü ïîðÿäîê ýòîãî óðàâíåíèÿ.33I. Ôóíêöèÿ G íå çàâèñèò îò y, y 0 , . . . , y (k−1) :G x, y (k) , y (k+1) , . . . , y (n) = 0. ýòîì ñëó÷àå ìîæíî ñäåëàòü çàìåíóz(x) = y (k) (x),òîãäà óðàâíåíèå (5) ñòàíåò ïîðÿäêà n − k :G x, z, z 0 , . . . , . . . , z (n−k) = 0.II.
Ãðóïïèðîâêà.Ýòîò âàðèàíò ïîäõîäèò, êîãäà óðàâíåíèå (5) ìîæíî ïðèâåñòè ê âèäódH x, y, y 0 , . . . , y (n−1) = 0,dxïîðÿäîê óìåíüøàåòñÿ íà 1:H x, y, y 0 , . . . , y (n−1) = C.III. Ôóíêöèÿ G îäíîðîäíàÿ ïî y ,ò.å. äëÿ ëþáîãî íåíóëåâîãî λ èìååò ìåñòî ðàâåíñòâîG x, λy, λy 0 , . . . , λy (n) = λδ G x, y, y 0 , . .
. , y (n)äëÿ íåêîòîðîãî ÷èñëà δ .1Òàê, åñëè âçÿòü λ = , òî (5) ìîæíî çàïèñàòü â âèäåyy 0 y 00y (n)G x, 1, , , . . . ,= 0.y yyÌîæíî ïðåäïîëîæèòü, ÷òî çàìåíày 0 (x)z(x) =y(x)ïðèâåä¼ò ê íóæíîìó ðåçóëüòàòó. Äåéñòâèòåëüíî, èìååìy 0 (x) = y(x)z(x),y 00 = y 0 z + yz 0 ,y 00y0= z + z0,yyy 00= z2 + z0.y34(∗)Ïî èíäóêöèè ïîêàæåì, ÷òîy (k)= Φk z, z 0 , . .
. , z (k−1) .yÄëÿ k = 1, 2 ýòî âûâåëè âûøå, ïóñòü äëÿ k = i ñïðàâåäëèâîy (i)= Φi z, z 0 , . . . , z (i−1) ,yòîãäàd (i)dy=y = ïî ïðåäïîëîæåíèþ èíäóêöèè =yΦi z, z 0 , . . . , z (i−1) =dxdxde i+1 z, z 0 , . . . , z (i) ,y 0 Φi z, z 0 , . . . , z (i−1) +y Φi z, z 0 , . . . , z (i−1) = y 0 Φi z, z 0 , . . . , z (i−1) +y Φdxïîëó÷àåì(i+1)y (i)y0e i+1 z, z 0 , . . . , z (i) == Φi z, z 0 , .
. . , z (i−1) + Φyye i+1 z, z 0 , . . . , z (i) = Φi+1 z, z 0 , . . . , z (i) .zΦi z, z 0 , . . . , z (i−1) + Φ ðåçóëüòàòå, (*) ïðè òàêîé çàìåíå ñòàíåòG x, 1, z, z 2 + z 0 , . . . , Φn z, z 0 , . . . , z (n−1)= 0,ýòî ìîæíî ïåðåîáîçíà÷èòüW x, z, z 0 , . . . , z (n−1) = 0,ïîëó÷èëè óðàâíåíèå íà ïîðÿäîê íèæå.IV. Ôóíêöèÿ G íå çàâèñèò îò x â ÿâíîì âèäå. ñëó÷àå, åñëè óðàâíåíèå (5) èìååò âèäG y, y 0 , .
. . , y (n) = 0,ìîæíî ñäåëàòü çàìåíóòîãäày 0 = p(y),y 00 (x) = (p(y(x)))0x = p0y (y(x))yx0 (x) = p0 p.Ïî èíäóêöèè ïîêàæåì, ÷òîy (k) = Φk p, p0 , . . . , p(k−1) .Äëÿ k = 1, 2 ýòî âûâåëè âûøå, ïóñòü äëÿ k = i ñïðàâåäëèâîy (i) = Φi p, p0 , . . . , p(i−1) ,35òîãäàdy (i) (x)dy== ïî ïðåäïîëîæåíèþ èíäóêöèè =Φk p, p0 , . . . , p(k−1) =dxdx dp(y(x)) dp0 (y(x))∂∂Φk p, p0 , . .
. , p(k−1)+ 0 Φk p, p0 , . . . , p(k−1)+ ···+∂pdx∂pdx 0 (k−1) (y(x))∂∂0(k−1)0(k−1) dpp p+=Φp,p,...,pΦp,p,...,pkk∂p(k−1)dx∂p 00 (k)∂∂0(i)0(k−1)0(k−1).pp+···+pp=Φp,p,...,pΦp,p,...,pΦp,p,...,pi+1kk∂p0∂p(k−1)(i+1)(x) ðåçóëüòàòå, (5) ïðè òàêîé çàìåíå ñòàíåòG y, p, pp0 , . . . , Φn p, p0 , . . . , p(n−1)= 0,ýòî ìîæíî ïåðåîáîçíà÷èòüW y, p, p0 , . . . , p(n−1) = 0,ïîëó÷èëè óðàâíåíèå íà ïîðÿäîê íèæå.V. Çàìåíà Ýéëåðà. ñëó÷àå, åñëè óðàâíåíèå (5) èìååò âèäG y, xy 0 , x2 y 00 , .
. . , xn y (n) = 0,çàìåíèì ïåðìåííóþ x íà íîâóþ t : x = ϕ(t), òîãäà y(x) = y(ϕ(t)), ýòî ìîæíî îáîçíà÷èòü ÷åðåç íîâóþ ôóíêöèþ u(t):y(x) = y(ϕ(t)) = u(t) = u(t(x)).Ïîïðîáóåì âçÿòü òàêóþ ôóíêöèþ ϕ, ÷òîáû ìîæíî áûëî çàìåíèòü xy 0 íà u0 , äëÿ ýòîãîðàñïèøåì:dy(x),xy 0 (x) = ϕ(t)dxdy(ϕ(t))dy(ϕ) ϕ(t)dy(x) ϕ(t)du(t)u0 (t) ====,dtdtdϕ dtdx dtϕ(t). Âñå ðåøåíèÿ ýòîãî óðàâdttíåíèÿ çàïèñûâàþòñÿ êàê ϕ(t) = Ce , ïîýòîìó ìîæíî âûáðàòü ñëåäóþùóþ çàìåíó:åñëè x > 0, òî x = et ,åñëè x < 0, òî x = −et .Ðàññìîòðèì ïåðâûé ñëó÷àé, x > 0, çàìåíà x = et , t = lnx, òîãäàçíà÷èò, íàäî ïîäîáðàòü ôóíêöèþ ϕ òàê, ÷òîáû ϕ(t) =y(x) = y(ϕ(t)) = u(t) = u(lnx).xy 0 (x) = xdu(t(x))du(t) dlnxdu(t) 1d=x=x= u0 (t) = u,dxdt dxdt xdt36dx y (x) = xdx2 002u0 (t(x))x= x2u00 (t)t0 (x)x − u0 (t)=x2ddu (t) − u (t) = (u0 − u) =dtdt000d− 1 u.dtÏî èíäóêöèè ïîêàæåì, ÷òîk (k)x y0= Φk u, u , .
. . , u(k)d=dt ddd−1− 2 ...− k + 1 u.dtdtdtÄëÿ k = 1, 2 ýòî âûâåëè âûøå, ïóñòü äëÿ k = i ñïðàâåäëèâî d dddi (i)xy =−1− 2 ...− i + 1 u,dt dtdtdtòîãäàx y(x) = ïî ïðåäïîëîæåíèþ èíäóêöèè = d !ddd−1−2...−i+1uddt dtdtdt=xi+1idxx ddd− 2 . . . dtd − i + 1 u0 (t)t0 (x)xi − dtd dtd − 1 dtd − 2 . . . dtd − i + 1 uixi−1i+1 dt dt − 1dtx=2ix ddd dddd d0−1− 2 ...− i + 1 u (t)−i−1− 2 ...−i+1 u=dt dtdtdtdt dtdtdt ddd d−1− 2 ...− i + 1 (u0 − iu) =dt dtdtdt d dddd−1− 2 ...−i+1− i u = Φi+1 u, u0 , . .
. , u(i+1) .dt dtdtdtdti+1 (i+1)Äîêàçàëè øàã èíäóêöèè. ðåçóëüòàòå, (5) ïðè òàêîé çàìåíå ñòàíåòG u, u0 , u00 − u0 , . . . , Φn u, u0 , . . . , u(n)ýòî ìîæíî ïåðåîáîçíà÷èòü= 0,W u, u0 , . . . , u(n) = 0,ïîëó÷èëè óðàâíåíèå âèäà IV.VI. Ôóíêöèÿ G îáîáù¼ííî-îäíîðîäíàÿ,ò.å. äëÿ ëþáîãî íåíóëåâîãî λ èìååò ìåñòî ðàâåíñòâîG λx, λα y, λα−1 y 0 , λα−2 y 00 , . . .
, λα−n y (n) = λδ G x, y, y 0 , . . . , y (n)äëÿ íåêîòîðîãî ÷èñëà δ .37Òàê, åñëè âçÿòü λ =1, òî (5) ìîæíî çàïèñàòü â âèäåxyy0y 00y (n)G 1, α , α−1 , α−2 , . . . , α−n = 0.x xxx(∗∗)Ìîæíî ïðåäïîëîæèòü, ÷òî çàìåíàz(x) =y(x)xαïðèâåä¼ò ê íóæíîìó ðåçóëüòàòó. Äåéñòâèòåëüíî, èìååìy(x) = xα z(x),y 0 = αxα−1 z + xα z 0 ,y0= αz + xz 0 .xα−1Ïî èíäóêöèè ïîêàæåì, ÷òîy (k)= Φk z, xz 0 , .
. . , xk z (k) .α−kxÄëÿ k = 1 ýòî âûâåëè âûøå, ïóñòü äëÿ k = i ñïðàâåäëèâîy (i)= Φi z, xz 0 , . . . , xi z (i) ,α−ixòîãäà1d (i)y (i+1)= α−i−1 y = ïî ïðåäïîëîæåíèþ èíäóêöèè =xα−i−1xdx1dα−i0i (i)xΦz,xz,...,xz=ixα−i−1 dx1α−i−10i (i)α−i d0i (i)(α − 1)xΦi z, xz , . . . , x zΦi z, xz , . . . , x z=+xxα−i−1dxd(α − 1)Φi z, xz 0 , . . . , xi z (i) + x Φi z, xz 0 , . . . , xi z (i) =dxd∂d∂d i (i)∂0i (i)0(α−1)Φi z, xz , . .
. , x z +xΦi z +Φi (xz ) + · · · +Φi (x z ) =∂z dx∂(xz 0 ) dx∂(xi z (i) ) dxe 1z0 + Φe 2 (z 0 + xz 00 ) + · · · + Φe i (ixi−1 z (i) + xi z (i+1) ) =(α − 1)Φi z, xz 0 , . . . , xi z (i) + x Φiii0i (i)10i+1(i+1)20i+1 (i+1)ee+Φz+. . .(α−1)Φi z, xz , . . . , x z + Φzi+1 z, xz , . . . , xi+1 z, xz , . . . , xi0i+1(i+1)0i+1(i+1)e= Φi+1 z, xz , . . .
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