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x<br />
= + + + .<br />
2<br />
2<br />
5<br />
y x c1 c2x c3<br />
x − 3 y′′ + y′<br />
= 0.<br />
Ð î ç â ’ ÿ ç à í í ÿ. гâíÿííÿ íå ì³ñòèòü ÿâíî ôóíêö³¿ y.<br />
Ïîêëàäåìî y′ = p, îòðèìàºìî y′′ = dp/<br />
dx, ³ ï³ñëÿ ï³äñòàíîâêè<br />
ó ïî÷àòêîâå ð³âíÿííÿ âîíî ïåðåòâîðþºòüñÿ íà ð³âíÿííÿ<br />
1-ãî ïîðÿäêó:<br />
dp<br />
( x − 3)<br />
+ p = 0 .<br />
dx<br />
³äîêðåìëþþ÷è çì³íí³ é ³íòåãðóþ÷è, çíàéäåìî<br />
dp dx<br />
+ = 0 ; ln p + ln x− 3 = ln c ( c > 0) ⇒ p( x − 3) =<br />
p x − 3<br />
Ïðèêëàä 11.8.2. Ðîçâ’ÿçàòè ð³âíÿííÿ ( )<br />
= c ⇒ p( x − 3) = ± c = c .<br />
1<br />
Çàì³íþþ÷è äîïîì³æíó çì³ííó p ÷åðåç dy<br />
dx , îòðèìàºìî<br />
ð³âíÿííÿ ( x − 3) dy = c1<br />
, ðîçâ’ÿçóþ÷è ÿêå, çíàéäåìî øóêàíèé<br />
dx<br />
çàãàëüíèé ³íòåãðàë:<br />
cdx<br />
= ⇒ = ln − 3 +<br />
x − 3<br />
1<br />
dy y c1 x c2<br />
11.9. ÇÀÑÒÎÑÓÂÀÍÍß ÄÈÔÅÐÅÍÖ²ÀËÜÍÈÕ<br />
вÂÍßÍÜ Â ÅÊÎÍÎֲ̲<br />
Íà ïî÷àòêó ï. 11.1 ìè ðîçãëÿíóëè ìàòåìàòè÷íó ìîäåëü<br />
åêîíîì³÷íîãî çì³ñòó, ÿêà áóëà çâåäåíà äî äèôåðåíö³àëüíîãî<br />
ð³âíÿííÿ.  öüîìó ïóíêò³ ïðîäîâæèìî ðîçãëÿäàííÿ çàäà÷<br />
òàêîãî õàðàêòåðó.<br />
11.9.1. Ìîäåëü Åâàíñà<br />
Ðîçãëÿíåìî ðèíîê îäíîãî òîâàðó. Íåõàé d(t), s(t), p(t) —<br />
â³äïîâ³äíî ôóíêö³¿ ïîïèòó, ïðîïîçèö³¿ ³ ö³íè öüîãî òîâàðó.<br />
Ìàòåìàòè÷íà ìîäåëü ð³âíîâàæíî¿ ö³íè áàçóºòüñÿ íà òàê³é<br />
.<br />
îñíîâí³é ã³ïîòåç³: ïðèð³ñò ö³íè çà ïðîì³æîê ÷àñó ∆t ïðÿìî<br />
ïðîïîðö³éíèé ð³çíèö³ ì³æ ïîïèòîì ³ ïðîïîçèö³ºþ, òîáòî<br />
∆ p = γ( d −s) ∆t, γ > 0. (11.9.1)<br />
Ðîçä³ëèìî îáèäâ³ ÷àñòèíè ð³âíîñò³ (11.9.1) íà ∆t ≠ 0 ³<br />
ïåðåéäåìî äî ãðàíèö³ ïðè ∆t → 0.<br />
Ó ðåçóëüòàò³ îòðèìàºìî äèôåðåíö³àëüíå ð³âíÿííÿ.<br />
dp<br />
=γ( d() t − p()<br />
t)<br />
. (11.9.2)<br />
dt<br />
Ïðèêëàä 11.9.1. Äîñë³äèòè ìîäåëü Åâàíñà, ïðèïóñêàþ÷è,<br />
ùî d ³ s â³äíîñíî ö³íè ð — ë³í³éí³ ôóíêö³¿:<br />
( ) , s( p)<br />
d p = a − b p = a + b p. (11.9.3)<br />
1 1 2 2<br />
Óñ³ êîíñòàíòè, ÿê³ âõîäÿòü ó ôîðìóëè (11.9.3), ââàæàþòüñÿ<br />
äîäàòíèìè ñòàëèìè. Ïðèïóñêàºòüñÿ òàêîæ, ùî a 1 > a 2<br />
(ïðè äîñòàòíüî ìàë³é ö³í³ ïîïèò ïåðåâèùóº ïðîïîçèö³þ) ³<br />
ùî<br />
p(0) = p 0 . (11.9.4)<br />
Ð î ç â ’ ÿ ç à í í ÿ. гâíÿííÿ (11.9.2) ç óðàõóâàííÿì (11.9.3)<br />
çàïèøåìî ó âèãëÿä³:<br />
dp<br />
=−γ( −α+β p)<br />
, (11.9.5)<br />
dt<br />
äå α = a 1 – a 2 , β = b 1 + b 2 >0.<br />
Äèôåðåíö³àëüíå ð³âíÿííÿ (11.9.5) º ð³âíÿííÿì ç â³äîêðåìëþâàíèìè<br />
çì³ííèìè. Çã³äíî ç ï. 11.4.2 çîáðàçèìî éîãî<br />
çàãàëüíèé ðîçâ’ÿçîê ó âèãëÿä³:<br />
−γβt<br />
α+ Ce<br />
p()<br />
t = .<br />
β<br />
Âðàõîâóþ÷è ïî÷àòêîâó óìîâó (11.9.4), áóäåìî ìàòè<br />
α<br />
p t p e e<br />
β<br />
−γβt<br />
−γβt<br />
() =<br />
0<br />
+ (1 − )<br />
. (11.9.6)<br />
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