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ÂÏÐÀÂÈ<br />
Ðîçâ’ÿçàòè ð³âíÿííÿ:<br />
dy 2xy<br />
11.15. = ;<br />
2 2<br />
dx x + y<br />
dy<br />
dx<br />
y<br />
x<br />
11.16. = ( + y − x)<br />
Âêàç³âêà ln y − ln x = ln y , y > 0, x > 0;<br />
x<br />
11.17.<br />
2 2<br />
y x xy<br />
1 ln ln ;<br />
+ dy dy<br />
;<br />
dx<br />
= dx<br />
11.18. ( y + x ) dy = ( y −x ) dx ;<br />
y + y −x dx − xdy = 0;<br />
2 2<br />
11.19. ( )<br />
2 2 2<br />
y xy x dx x dy<br />
11.20. ( )<br />
− + − = 0;<br />
⎛<br />
11.21. cos y ⎞ y<br />
⎜x − y dx + xcos dy = 0<br />
x<br />
⎟<br />
.<br />
⎝<br />
⎠ x<br />
11.5. ˲ͲÉͲ ÄÈÔÅÐÅÍÖ²ÀËÜͲ<br />
вÂÍßÍÍß ÏÅÐØÎÃÎ ÏÎÐßÄÊÓ<br />
11.5.1. Îçíà÷åííÿ<br />
гâíÿííÿ âèãëÿäó<br />
y′ + P( x) y = Q( x)( x∈ ( a, b))<br />
, (11.5.1)<br />
äå ôóíêö³¿ P(x), Q(x) — â³äîì³ òà íåïåðåðâí³ íà ³íòåðâàë³ (a, b),<br />
à øóêàíà ôóíêö³ÿ y º íåïåðåðâíî äèôåðåíö³éîâíà íà íüîìó, íàçèâàºòüñÿ<br />
ë³í³éíèì äèôåðåíö³àëüíèì ïåðøîãî ïîðÿäêó.<br />
Äëÿ ðîçâ’ÿçàííÿ ð³âíÿííÿ (11.5.1) âèêîðèñòîâóþòü äâà<br />
ìåòîäè: ìåòîä âàð³àö³¿ äîâ³ëüíî¿ ñòàëî¿ òà ìåòîä Áåðíóëë³<br />
– Ôóð’º. Ïåðøèì ìåòîäîì ð³âíÿííÿ (11.5.1) ³íòåãðóºòüñÿ<br />
òàêèì ÷èíîì. Ðîçãëÿíåìî ë³í³éíå îäíîð³äíå ð³âíÿííÿ<br />
y′ + P( x) y = 0, ÿêå îòðèìàíî ç (11.5.1) ïðè Q(x) ≡ 0. Â öüîìó<br />
ð³âíÿíí³ çì³íí³ ëåãêî â³äîêðåìëþþòüñÿ.  ðåçóëüòàò³ ìàºìî<br />
dy<br />
=−Pxy ( ) ⇒ ln y =− ∫ Pxdx ( ) + lnc,<br />
dx<br />
äå äëÿ çðó÷íîñò³ äîâ³ëüíà ñòàëà çîáðàæåíà ÿê ln ⏐c⏐. ϳñëÿ<br />
ïîòåíö³þâàííÿ îñòàííüî¿ ð³âíîñò³ îòðèìàºìî çàãàëüíèé ðîçâ’ÿçîê<br />
îäíîð³äíîãî ë³í³éíîãî ð³âíÿííÿ ïåðøîãî ïîðÿäêó:<br />
( )<br />
y x<br />
P( x)<br />
dx<br />
= ce −∫<br />
. (11.5.2)<br />
Áóäåìî òåïåð ââàæàòè â (11.5.2) ñ íåâ³äîìîþ ôóíêö³ºþ<br />
â³ä x ³ âèçíà÷èìî ¿¿ ç óìîâè çàäîâîëåííÿ ðîçâ’ÿçêó<br />
( )<br />
y x<br />
P( x)<br />
dx<br />
= c( x)<br />
e − ∫<br />
(11.5.3)<br />
ð³âíÿííÿ (11.5.1). Äèôåðåíö³þþ÷è (11.5.3) ³ ï³äñòàâëÿþ÷è<br />
äî (11.5.1), îòðèìàºìî<br />
dc − P( x) dx P( x) dx P( x)<br />
dx<br />
e<br />
∫<br />
−<br />
c x P x e<br />
∫<br />
−<br />
c x P x e<br />
∫ Q x<br />
dx<br />
²íòåãðóþ÷è, ìàºìî<br />
( ) ( ) ( ) ( ) ( )<br />
− + = ⇒<br />
( ) ( )<br />
P( x)<br />
dx<br />
⇒ c′<br />
x = Q x e ∫ .<br />
P( x)<br />
dx<br />
c( x) = Q( x)<br />
e<br />
∫<br />
∫ dx + c .<br />
1<br />
ϳäñòàâëÿþ÷è äî (11.5.3) âèðàç äëÿ c(x), îòðèìàºìî<br />
P( x) dx P( x) dx P( x)<br />
dx<br />
y x = c e<br />
∫<br />
+ e<br />
∫<br />
Q x e<br />
∫<br />
∫ dx. (11.5.4)<br />
−<br />
−<br />
( ) 1<br />
( )<br />
Ïðèêëàä 11.5.1. Çíàéòè çàãàëüíèé ðîçâ’ÿçîê ð³âíÿííÿ<br />
dy<br />
dx<br />
y<br />
x<br />
2<br />
− = x . (11.5.5)<br />
Ð î ç â ’ ÿ ç à í í ÿ. Ðîçãëÿíåìî ñïî÷àòêó â³äïîâ³äíå ë³í³éíå<br />
îäíîð³äíå ð³âíÿííÿ<br />
dy y<br />
− = 0 . (11.5.6)<br />
dx x<br />
dy dx<br />
 öüîìó ð³âíÿíí³ çì³íí³ â³äîêðåìëþþòüñÿ: = .<br />
y x<br />
²íòåãðóþ÷è,<br />
îòðèìàºìî<br />
y = cx. (11.5.7)<br />
420 421
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