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Òåîðåìà Êîø³ ðîçâ’ÿçóº á³ëüø ãëîáàëüíó ïðîáëåìó í³æ<br />
íàâåäåíà âèùå çàäà÷à. Âîíà äຠìîæëèâ³ñòü çà âèäîì äèôåðåíö³àëüíîãî<br />
ð³âíÿííÿ âèð³øóâàòè ïèòàííÿ ïðî ³ñíóâàííÿ<br />
òà ºäèí³ñòü éîãî ðîçâ’ÿçêó. Öå îñîáëèâî âàæëèâî â òèõ<br />
âèïàäêàõ, êîëè çàçäàëåã³äü íåâ³äîìî, ÷è ìຠäàíå ð³âíÿííÿ<br />
ðîçâ’ÿçîê.<br />
11.4.1. Ïîëå íàïðÿì³â (³çîêë³íè)<br />
Íàéá³ëüø íàî÷íî ãåîìåòðè÷íèé çì³ñò äèôåðåíö³àëüíîãî<br />
ð³âíÿííÿ y′ = f( x, y)<br />
âäàºòüñÿ ïðî³ëþñòðóâàòè çà äîïîìîãîþ<br />
ïîëÿ íàïðÿì³â, ÿêå õàðàêòåðèçóºòüñÿ ð³âíÿííÿì<br />
f(x, y) =k. (11.4.4)<br />
Âèáåðåìî êîíêðåòíó òî÷êó M 0 (x 0 , y 0 ) ç îáëàñò³ D âèçíà-<br />
÷åííÿ ôóíêö³¿ f(x, y) ³ îá÷èñëèìî f(x 0 , y 0 ). ×èñëî f(x 0 , y 0 ) ó<br />
â³äïîâ³äíîñò³ äî ð³âíîñò³ (11.4.2) ÿâëÿº ñîáîþ êóòîâèé êîåô³ö³ºíò<br />
äîòè÷íî¿ äî ³íòåãðàëüíî¿ êðèâî¿ ð³âíÿííÿ (11.4.2).<br />
Ïîáóäóºìî â òî÷ö³ (x 0 , y 0 ) íàïðÿìëåíó äîòè÷íó ó âèãëÿä³<br />
íåâåëèêîãî â³äð³çêà. Ïðîâ³âøè ïîáóäîâó äëÿ óñ³õ òî÷îê<br />
îáëàñò³ D, îòðèìàºìî òàê çâàíå ïîëå íàïðÿì³â (ðèñ. 11.3).<br />
Ðèñ. 11.2<br />
ßê ó íàâåäåíîìó ïðèêëàä³, òàê ³ â ãëîáàëüíîìó ñåíñ³<br />
òåîðåìà Êîø³ çà ïåâíèõ óìîâ ñòâåðäæóº, ùî ÷åðåç êîæíó<br />
âíóòð³øíþ òî÷êó (x 0 , y 0 ) îáëàñò³ D (â ÿê³é çàäàíà ôóíêö³ÿ<br />
f(x, y)) ïðîõîäèòü ºäèíà ³íòåãðàëüíà êðèâà. Î÷åâèäíî, ùî<br />
ïðè öüîìó ð³âíÿííÿ (11.4.2) ìຠíåñê³í÷åííó ìíîæèíó ðîçâ’ÿçê³â.<br />
Ðîçøóê ðîçâ’ÿçêó ð³âíÿííÿ (11.4.2), ÿêèé çàäîâîëüíÿº<br />
ïî÷àòêîâó óìîâó (11.4.3), íàçèâàºòüñÿ çàäà÷åþ Êîø³. Ç ãåîìåòðè÷íî¿<br />
òî÷êè çîðó, ðîçâ’ÿçàòè çàäà÷ó Êîø³ îçíà÷àº, ùî<br />
³ç ìíîæèíè ³íòåãðàëüíèõ êðèâèõ òðåáà âèä³ëèòè òó, ÿêà<br />
ïðîõîäèòü ÷åðåç çàäàíó òî÷êó (x 0 , y 0 ) ïëîùèíè Oxy.<br />
Òî÷êè ïëîùèíè, ÷åðåç ÿê³ ïðîõîäèòü á³ëüø îäí³º¿ ³íòåãðàëüíî¿<br />
êðèâî¿ àáî íå ïðîõîäèòü æîäíî¿, íàçèâàþòüñÿ îñîáëèâèìè<br />
òî÷êàìè äàíîãî ð³âíÿííÿ.<br />
Ðèñ 11.3<br />
Òàêèì ÷èíîì, ãåîìåòðè÷íî ð³âíÿííÿ (11.4.4) çàäຠïîëå<br />
íàïðÿì³â, äîòè÷íèõ äî ãðàô³ê³â ðîçâ’ÿçê³â öüîãî ð³âíÿííÿ,<br />
òîáòî â êîæí³é òî÷ö³ îáëàñò³ D íàïðÿìîê äîòè÷íî¿ äî ³íòåãðàëüíî¿<br />
êðèâî¿ çá³ãàºòüñÿ ç íàïðÿìîì ïîëÿ â ö³é òî÷ö³.<br />
Ïîáóäóâàâøè íà ïëîùèí³ ïîëå íàïðÿì³â äàíîãî äèôåðåíö³àëüíîãî<br />
ð³âíÿííÿ, ìîæíà íàáëèæåíî ïîáóäóâàòè ³íòåãðàëüí³<br />
êðèâ³.<br />
2 2<br />
Ïðèêëàä 11.4.1. Ðîçãëÿíåìî ð³âíÿííÿ y′ = x + y . Äëÿ<br />
åôåêòèâíîãî âèçíà÷åííÿ ïîëÿ íàïðÿì³â âèêîðèñòàºìî êðèâ³,<br />
âçäîâæ ÿêèõ y′ = c . Ö³ êðèâ³ ïðèéíÿòî çâàòè ³çîêë³íàìè.<br />
Ó öüîìó âèïàäêó ³çîêë³íè çîáðàæóþòü ñîáîþ ñ³ìåéñòâî<br />
x 2 + y 2 = c c ≥ 0 ç öåíòðîì íà ïî÷àòêó<br />
êîíöåíòðè÷íèõ ê³ë ( )<br />
410 411
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