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êîîðäèíàò (ðèñ. 11.4). Äëÿ íàî÷íîñò³ ïîêëàäåìî òàê³ çíà-<br />
÷åííÿ c: c = 0, c = 3/3, c =1, c = 3 . Ïîçíà÷èìî ÷åðåç ϕ<br />
êóò íàõèëó äîòè÷íî¿. Òîä³ ïðè c = 0 (tg ϕ =0)⇒ ϕ =0,<br />
c = 3/3 ⇒ ϕ = π/6, c =1⇒ ϕ = π/4, c = 3 ⇒ ϕ = π/3. Ó âèïàäêó<br />
c = 0 êîëî âèðîäæóºòüñÿ ó òî÷êó.<br />
Ðèñ 11.4<br />
Ìàþ÷è ïîëå íàïðÿì³â, íåâàæêî ïîáóäóâàòè ³íòåãðàëüí³<br />
êðèâ³. Äëÿ íàáëèæåíî¿ ïîáóäîâè êîíêðåòíî¿ ³íòåãðàëüíî¿<br />
êðèâî¿ íåîáõ³äíî çàäàòè òî÷êó, ÷åðåç ÿêó âîíà ïðîõîäèòü.<br />
Äëÿ âèçíà÷åíîñò³ íåõàé öå áóäå òî÷êà Î(0,0). Ïîáóäîâàíà<br />
êðèâà L º íàáëèæåíèì ÷àñòèííèì ðîçâ’ÿçêîì ðîçãëÿíóòîãî<br />
ð³âíÿííÿ.<br />
Ñôîðìóëþºìî òåîðåìó Êîø³ äëÿ ð³âíÿííÿ n-ãî ïîðÿäêó,<br />
ðîçâ’ÿçàíîãî â³äíîñíî ñòàðøî¿ ïîõ³äíî¿<br />
ç ïî÷àòêîâèìè óìîâàìè<br />
( , , ′, ′′, , )<br />
( n) ( n 1)<br />
y = f x y y y K y −<br />
(11.4.5)<br />
( n−1) ( n−1)<br />
yx ( ) = y, y′ ( x) = y′<br />
, K , y ( x)<br />
= y . (11.4.6)<br />
0 0 0 0 0 0<br />
Òåîðåìà 11.4.2. ßêùî ôóíêö³ÿ<br />
( n 1)<br />
( , , , , , y )<br />
−<br />
f x y y′ y′′ K , ùî<br />
( n 1)<br />
çàëåæèòü â³ä n + 1 çì³ííèõ xyy , , ′, y′′ , K , y − , âèçíà÷åíà ³<br />
íåïåðåðâíà â äåÿê³é (n + 1)-âèì³ðí³é â³äêðèò³é îáëàñò³ D<br />
ðàçîì ç ÷àñòèííèìè ïîõ³äíèìè ∂ f<br />
, ∂ f f<br />
, K ,<br />
∂<br />
( n 1<br />
∂y ∂y′<br />
)<br />
∂y − , òî äëÿ<br />
( )<br />
( )<br />
1<br />
âñÿêî¿ òî÷êè , , , , ,<br />
n<br />
x0 y0 y′ 0<br />
y′′ 0<br />
K y −<br />
0 , ÿêà íàëåæèòü îáëàñò³ D,<br />
â äåÿêîìó îêîë³ òî÷êè x = x 0 ³ñíóº ºäèíèé ðîçâ’ÿçîê ð³âíÿííÿ<br />
(11.4.5), ÿêèé çàäîâîëüíÿº ïî÷àòêîâ³ óìîâè (11.4.6).<br />
Öþ òåîðåìó Êîø³ ïîäàìî òàêîæ áåç äîâåäåííÿ.<br />
Äëÿ äèôåðåíö³àëüíèõ ð³âíÿíü n-ãî ïîðÿäêó òåæ ââîäÿòü<br />
ïîíÿòòÿ çàãàëüíîãî ðîçâ’ÿçêó. Àëå ó öüîìó âèïàäêó â³í<br />
ì³ñòèòü n äîâ³ëüíèõ ñòàëèõ.<br />
11.4.2. Êëàñè íåë³í³éíèõ äèôåðåíö³àëüíèõ ð³âíÿíü<br />
ïåðøîãî ïîðÿäêó, ÿê³ ðîçâ’ÿçóþòüñÿ ó êâàäðàòóðàõ<br />
Çàãàëüíîãî ìåòîäó çíàõîäæåííÿ ðîçâ’ÿçê³â äèôåðåíö³àëüíèõ<br />
ð³âíÿíü ïåðøîãî ïîðÿäêó íå ³ñíóº. Çâè÷àéíî ðîçãëÿäàþòüñÿ<br />
îêðåì³ òèïè ð³âíÿíü, ³ äëÿ êîæíîãî ç íèõ çíàõîäÿòü<br />
ñâ³é ñïîñ³á ðîçâ’ÿçàííÿ.<br />
1.4.2.1. гâíÿííÿ ç â³äîêðåìëåíèìè òà â³äîêðåìëþâàíèìè<br />
çì³ííèìè<br />
Äèôåðåíö³àëüíå ð³âíÿííÿ<br />
M( x) dx + N( y) dy = 0<br />
(11.4.7)<br />
íàçèâàºòüñÿ ð³âíÿííÿì ç â³äîêðåìëåíèìè çì³ííèìè. Çàãàëüíèé<br />
³íòåãðàë çíàõîäèòüñÿ çà ôîðìóëîþ<br />
∫M( x) dx + ∫ N( y)<br />
dy = c . (11.4.8)<br />
гâíÿííÿ âèãëÿäó<br />
M ( x) N ( y) dx + M ( x) N ( y) dy = 0 (11.4.9)<br />
1 1 2 2<br />
íàçèâàþòüñÿ ð³âíÿííÿìè ç â³äîêðåìëþâàíèìè çì³ííèìè.<br />
Âîíè ìîæóòü áóòè ïðèâåäåí³ äî ð³âíÿíü òèïó (11.4.7) øëÿõîì<br />
ä³ëåííÿ îáîõ ÷àñòèí íà âèðàç N1( y) M2( x ):<br />
M1( x) N1( y)<br />
dx + dy = 0 .<br />
M ( x) N ( y)<br />
(11.4.10)<br />
2 2<br />
412 413
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