Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
y′ ÷.í. = e x (2Ax +Ax 2 ),<br />
y′′ ÷.í. =(2A +2Ax)e x +(2Ax + Ax 2 )e x =(Ax 2 +4Ax +2A)e x .<br />
ϳäñòàâëÿþ÷è y ÷.í. , y′ ÷.í. ³ y′′ ÷.í. äî ð³âíÿííÿ (11.7.8) ³ ñêîðî÷óþ÷è<br />
íà e x ≠ 0, îòðèìàºìî 2Ax 2 +4Ax +2A –4Ax –2Ax 2 =1<br />
1<br />
àáî 2A = 1, çâ³äêè A = . Òàêèì ÷èíîì, ÷àñòèííèé ðîçâ’ÿçîê<br />
íåîäíîð³äíîãî ð³âíÿííÿ (11.7.8) ìàº<br />
2<br />
âèãëÿä:<br />
à çàãàëüíèé<br />
1 x<br />
= xe ,<br />
2<br />
y ÷.í.<br />
2<br />
x<br />
1 2 x<br />
y ç.í. = e ( c1 + xc2) + x e .<br />
2<br />
II. Ïðàâà ÷àñòèíà ð³âíÿííÿ<br />
αx<br />
fx ( ) = e ( Pn( x)cos β x+ Qm( x)sinβ x)<br />
.<br />
Òóò P n (x) ³ Q m (x) — ìíîãî÷ëåíè ñòåïåí³â â³äïîâ³äíî n ³<br />
m.  öüîìó âèïàäêó ìîæëèâ³ äâà âàð³àíòè:<br />
1) ÿêùî ÷èñëî α + iβ íå º êîðåíåì õàðàêòåðèñòè÷íîãî<br />
ð³âíÿííÿ, òî ÷àñòèííèé ðîçâ’ÿçîê ð³âíÿííÿ (11.7.3) ñë³ä<br />
øóêàòè ó âèãëÿä³<br />
y ÷.í. ( ( )cos ( )sin )<br />
x<br />
= Rl<br />
x β x + Ml<br />
x β x e α , (11.7.9)<br />
äå R l (x) ³ M l (x) — ìíîãî÷ëåíè ñòåïåíÿ l = max (n, m);<br />
2) ÿêùî ÷èñëî α + iβ º êîð³íü õàðàêòåðèñòè÷íîãî ð³âíÿííÿ,<br />
òî ÷àñòèííèé ðîçâ’ÿçîê ñë³ä øóêàòè ó âèãëÿä³<br />
y ÷.í. ( )<br />
x<br />
= x R ( x) cos β x + M ( x) sin β x e , l = max( m, n)<br />
. (11.7.10)<br />
l<br />
l<br />
Çàóâàæåííÿ. Äëÿ çàïîá³ãàííÿ ìîæëèâèõ ïîìèëîê â³äçíà÷èìî,<br />
ùî ñòðóêòóðà ÷àñòèííèõ ðîçâ’ÿçê³â çáåð³ãຠâèãëÿäè<br />
(11.7.9 – 11.7.10) ³ â òîìó âèïàäêó, ÿêùî îäèí ç ìíîãî-<br />
÷ëåí³â P n (x) àáî Q m (x) òîòîæíî äîð³âíþº íóëþ.<br />
Ïðèêëàä 11.7.7. Çíàéòè çàãàëüíèé ðîçâ’ÿçîê ð³âíÿííÿ<br />
y′′ + 2y′<br />
+ 5y = 2cosx. (11.7.11)<br />
Ð î ç â ’ ÿ ç à í í ÿ. Õàðàêòåðèñòè÷íå ð³âíÿííÿ k 2 +2k +5=0<br />
ìຠêîðåí³ k 1 =–1+2i, k 2 = –1– 2i. Îòæå, çàãàëüíèé ðîçâ’ÿçîê<br />
â³äïîâ³äíîãî îäíîð³äíîãî ð³âíÿííÿ ìຠâèãëÿä:<br />
−x<br />
y ç.o. e ( c cos 2x c sin 2x)<br />
= + .<br />
1 2<br />
Ó â³äïîâ³äíîñò³ äî ôîðìóëè (11.7.9) ÷àñòèííèé ðîçâ’ÿçîê<br />
íåîäíîð³äíîãî ð³âíÿííÿ øóêàºìî ó âèãëÿä³<br />
y ÷.í. = A cosx + B sinx.<br />
Äàë³, çíàõîäèìî ïîõ³äí³ ïåðøîãî ³ äðóãîãî ïîðÿäê³â ö³º¿<br />
ôóíêö³¿:<br />
y′ ÷.í. =− Asin x + Bcos<br />
x ,<br />
y′′ ÷.í. =–A cosx – B sinx.<br />
ϳäñòàâëÿþ÷è y ÷.í. , y′ ÷.í. ³ y′′ ÷.í. äî ð³âíÿííÿ (11.7.11), áóäåìî<br />
ìàòè<br />
−Acos x − Bsin x − 2Asin x + 2Bcos x + 5Acos x + 5Bsin x = 2 cos x.<br />
Ïðèð³âíþþ÷è êîåô³ö³ºíòè ïðè cos x ³ sinx îòðèìàºìî<br />
A + 2B+ 5A = 2; − B− 2A + 5B<br />
= 0,<br />
2 1<br />
çâ³äêè A = , B = .<br />
5 5<br />
Çàãàëüíèé ðîçâ’ÿçîê íåîäíîð³äíîãî ð³âíÿííÿ (11.7.11)<br />
òàêèé:<br />
−x<br />
2 1<br />
y ç.í. = y ç.o. + y ÷.í. = e ( c1 cos 2x + c2<br />
sin 2x)<br />
+ cos x + sin x.<br />
5 5<br />
Ïðèêëàä 11.7.8. Çíàéòè çàãàëüíèé ðîçâ’ÿçîê ð³âíÿííÿ<br />
y′′ + 4y = cos2x. (11.7.12)<br />
440 441
Change language