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Ñë³ä çàóâàæèòè, ùî ð³âíÿííÿ (11.4.9) – (11.4.10) áóäóòü<br />
åêâ³âàëåíòí³ ó ò³é îáëàñò³, äå ôóíêö³¿ M 2 (x) ³ N 2 (y) íå ïåðåòâîðþþòüñÿ<br />
äî íóëÿ. Òàêèì ÷èíîì, ó çàãàëüíîìó âèïàäêó<br />
ïðè ðîçãëÿä³ ð³âíÿííÿ (11.4.10) ðîçâ’ÿçîê ð³âíÿííÿ (11.4.9)<br />
ìîæå áóòè âòðà÷åíèì. Ó öüîìó âèïàäêó ìîæëèâ³ âòðà÷åí³<br />
ðîçâ’ÿçêè ïåðåâ³ðÿþòüñÿ áåçïîñåðåäíüî, øëÿõîì ï³äñòàíîâêè<br />
M 2 (x) =0 ³ N 2 (y) = 0 ó âèõ³äíå ð³âíÿííÿ.<br />
Ïðèêëàä 11.4.2. Ðîçâ’ÿçàòè ð³âíÿííÿ<br />
xdx + ydy = 0 . (11.4.11)<br />
Ð î ç â ’ ÿ ç à í í ÿ. ²íòåãðóþ÷è (11.4.11), îòðèìàºìî çàãàëüíèé<br />
ðîçâ’ÿçîê (³íòåãðàë)<br />
2 2<br />
x y<br />
+ = c1<br />
. (11.4.12)<br />
2 2<br />
Îñê³ëüêè ë³âà ÷àñòèíà ð³âíîñò³ (11.4.12) º íåâ³ä’ºìíîþ, òî<br />
êîíñòàíòà ñ 1 òåæ íåâ³ä’ºìíà. Ïîçíà÷èìî 2ñ 1 ÷åðåç ñ 2 , áóäåìî<br />
ìàòè<br />
2 2 2<br />
x + y = c .<br />
Îòæå, ³íòåãðàëüí³ êðèâ³ ðîçãëÿíóòîãî ð³âíÿííÿ ÿâëÿþòü<br />
ñîáîþ êîíöåíòðè÷í³ êîëà ç öåíòðîì íà ïî÷àòêó êîîðäèíàò<br />
³ ðàä³óñîì ñ.<br />
Ïðèêëàä 11.4.3. Ðîçâ’ÿçàòè ð³âíÿííÿ<br />
ydx + xdy = 0 . (11.4.13)<br />
Ð î ç â ’ ÿ ç à í í ÿ. Öå ð³âíÿííÿ ç â³äîêðåìëþâàíèìè çì³ííèìè.<br />
ijëèìî îáèäâ³ ÷àñòè íà xy. Îòðèìàºìî ð³âíÿííÿ<br />
dx dy<br />
+ = 0<br />
x y , (11.4.14)<br />
ÿêå íååêâ³âàëåíòíå âèõ³äíîìó ð³âíÿííþ â îáëàñò³, ùî ì³ñòèòü<br />
ïî÷àòîê êîîðäèíàò. Çàãàëüíèé ðîçâ’ÿçîê ð³âíÿííÿ<br />
(11.1.14) çíàõîäèòüñÿ øëÿõîì ³íòåãðóâàííÿ:<br />
ϳñëÿ ïîòåíö³þâàííÿ, áóäåìî ìàòè<br />
y =<br />
x<br />
àáî y =± .<br />
x<br />
Ïîêëàäåìî ±ñ 1 = ñ, îñòàòî÷íî îòðèìàºìî<br />
c 1<br />
c<br />
y = . (11.4.15)<br />
x<br />
Ãåîìåòðè÷íî çàãàëüíèé ðîçâ’ÿçîê ð³âíÿííÿ (11.4.14) ÿâëÿº<br />
ñîáîþ ñ³ìåéñòâî ð³âíîá³÷íèõ ã³ïåðáîë.<br />
Íåâàæêî ïîì³òèòè, ùî ïðè ä³ëåíí³ ð³âíÿííÿ (11.4.13) íà<br />
xy ìè âòðàòèëè ðîçâ’ÿçêè x = 0, y = 0, ÿê³ íå ìîæíà îòðèìàòè<br />
³ç çàãàëüíîãî ðîçâ’ÿçêó (11.4.15). Òóò ìè çíîâó òîðêíóëèñÿ<br />
ïèòàíü, ÿê³ ïîâ’ÿçàí³ ç òåîðåìîþ ³ñíóâàííÿ òà ºäèíîñò³<br />
äèôåðåíö³àëüíîãî ð³âíÿííÿ ïåðøîãî ïîðÿäêó (äèâ. òåîð.<br />
11.4.1).<br />
Ïðèêëàä 11.4.4. Ðîçâ’ÿçàòè ð³âíÿííÿ<br />
c 1<br />
2 2<br />
1 1 0<br />
− ydx+ − xdy= . (11.4.16)<br />
Ðîçâ’ÿçàííÿ. Îñê³ëüêè êîåô³ö³ºíòè ïðè dx ³ dy ìîæóòü<br />
áóòè çîáðàæåí³ ó âèãëÿä³ äîáóòêó ôóíêö³é, ùî çàëåæàòü<br />
ò³ëüêè â³ä x òà ò³ëüêè â³ä y, òî ð³âíÿííÿ (11.4.16) º<br />
ð³âíÿííÿì ç â³äîêðåìëþâàíèìè çì³ííèìè. ³äîêðåìèìî<br />
çì³íí³ ³ ç³íòåãðóºìî:<br />
dx dy<br />
∫ + ∫ = c .<br />
2 2<br />
1−x<br />
1−<br />
y<br />
Îòðèìàºìî çàãàëüíèé ³íòåãðàë<br />
arcsin x + arcsin y = c. (11.4.17)<br />
Ç (11.4.17) ìîæíà îäåðæàòè çàãàëüíèé ðîçâ’ÿçîê ó âèãëÿä³:<br />
y = sin( c − arcsin x)<br />
.<br />
ln x + ln y = ln c .<br />
1<br />
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