Ìiíiñòåðñòâî îñâiòè Óêðà íè Íàöiîíàëüíèé òåõíi íèé óíiâåðñèòåò Óêðà íè "Êè âñüêèé ïîëiòåõíi íèé iíñòèòóò" Êàôåäðà âèùî ìàòåìàòèêè N2 ÑÈÑÒÅÌÈ ÄÈÔÅÐÅÍÖIÀËÜÍÈÕ ÐIÂÍßÍÜ Íàâ àëüíî-ìåòîäè íèé ïîñiáíèê Êè â 999
Áîðèñåíêî Ñ.Ä., Äóäêií Ì.. Ñèñòåìè äèôåðåíöiàëüíèõ ðiâíÿíü: Íàâ. ïîñiáíèê. Ê.: ÍÒÓÓ "ÊÏI", 999. 25ñ. Ïîñiáíèê "Ñèñòåìè äèôåðåíöiàëüíèõ ðiâíÿíü"ìiñòèòü â ñîái ñòèñëèé òåîðåòè íèé ìàòåðiàë, çðàçêè ðîçâ'ÿçàííÿ çàäà òà çàâäàííÿ äëÿ ñàìîñòiéíîãî âèêîíàííÿ ç òåìè "Ëiíiéíi îäíîðiäíi i íåîäíîðiäíi ñèñòåìè äèôåðåíöiàëüíèõ ðiâíÿíü çi ñòàëèìè êîåôiöi- ¹íòàìè". Äëÿ ñòóäåíòiâ ôiçèêî-ìàòåìàòè íèõ ñïåöiàëüíîñòåé óíiâåðñèòåòiâ òà ïåäàãîãi íèõ iíñòèòóòiâ, ÿêi âèâ àþòü êóðñ "Äèôåðåíöiàëüíi ðiâíÿííÿ", çîêðåìà ðîçäië "Ñèñòåìè äèôåðåíöiàëüíèõ ðiâíÿíü". Ìàòåðiàë ìîæíà âèêîðèñòîâóâàòè íà çàíÿòòÿõ ç âèùî ìàòåìàòèêè íà òåõíi íèõ ôàêóëüòåòàõ óíiâåðñèòåòiâ òà iíñòèòóòiâ. Çà ðåäàêöi¹þ À.Ì.Ñàìîéëåíêà c ÍÒÓÓ "ÊÏI"
Âñòóï Êàíîíi íîþ íàçèâà¹òüñÿ ñèñòåìà ç k äèôåðåíöiàëüíèõ ðiâíÿíü, ùî ïîâ'ÿçóþòü íåçàëåæíó çìiííó t i k ôóíêöié y (t), y 2 (t),..., y k (t), ÿêà ðîçâ'ÿçàíà âiäíîñíî ñòàðøèõ ïîõiäíèõ öèõ ôóíêöié y p (t), yp 2 2 (t),...,yp k k (t), òîáòî ì๠âèãëÿä y (p i) i (t) = f i (t, y,...y (p ),..., y k,..., y (p k ) k ), i =, k, () äå p i ïîðÿäîê âiäïîâiäíî ïîõiäíî. èñëî n = p + p 2 +... + p k íàçèâàþòü ïîðÿäêîì ñèñòåìè. ßêùî p = p 2 =... = p k =, òî ñèñòåìó () íàçèâàþòü íîðìàëüíîþ ñèñòåìîþ ïåðøîãî ïîðÿäêó, òîáòî âîíà ì๠âèãëÿä y i = f i (x, y,...y n ), i =, n. (2) Ðîçâ'ÿçêîì ñèñòåìè (2) íà iíòåðâàëi (a, b) íàçèâàþòü ñóêóïíiñòü ôóíêöié y = ϕ (t),..., y n = ϕ n (t), íåïåðåðâíî äèôåðåíöiéîâíèõ íà (a, b), ÿêi îáåðòàþòü ðiâíÿííÿ öi¹ ñèñòåìè ó òîòîæíîñòi x (a, b). Ïiä ðîçâ'ÿçêîì çàäà i Êîøi ðîçóìiþòü çíàõîäæåííÿ ðîçâ'ÿçêiâ y (t),..., y n (t) ñèñòåìè (2), ÿêi çàäîâîëüíÿþòü ïî àòêîâi óìîâè y i (t ) = y i, i =, n, (3) äå yi, i =, n - çàäàíi èñëà. Òåîðåìà. Íåõàé ïðàâi àñòèíè f i, i =, n, íîðìàëüíî ñèñòåìè (2) âèçíà åíi â (n+)-âèìiðíié îáëàñòi D çìiííèõ x, y, y 2,...y n. ßêùî ó äåÿêîìó îêîëi òî êè M ç êîîðäèíàòàìè (x, y, y 2,..., y n) D ôóíêöi f ν ¹ íåïåðåðâíèìè i ìàþòü íåïåðåðâíi àñòèííi ïîõiäíi ν f y i çà çìiííèìè y, y 2,...,y n, òî çíàéäåòüñÿ òàêèé iíòåðâàë x h < x < x + h, â ÿêîìó iñíó¹ ¹äèíèé ðîçâ'ÿçîê ñèñòåìè (2), ÿêèé çàäîâîëüíÿ¹ ïî àòêîâi óìîâè (3). Çàãàëüíèì ðîçâ'ÿçêîì ñèñòåìè (2) íàçèâà¹òüñÿ ñóêóïíiñòü ôóíêöié y ν (x, C,..., C n ), ν =, n, (4) ÿêi çàëåæàòü âiä n äîâiëüíèõ ñòàëèõ i ïðè áóäü-ÿêèõ äîïóñòèìèõ çíà åííÿõ ñòàëèõ C i îáåðòàþòü ðiâíÿííÿ ñèñòåìè (2) ó òîòîæíîñòi.
 îáëàñòi, äå âèêîíóþòüñÿ óìîâè òåîðåìè, çà äîïîìîãîþ ôóíêöié (4) ìîæíà îòðèìàòè ðîçâ'ÿçîê áóäü-ÿêî çàäà i Êîøi. Íîðìàëüíà ñèñòåìà n-ãî ïîðÿäêó ó âèïàäêó, êîëè âîíà ¹ îäíîðiäíîþ, ì๠âèãëÿä ẋ = a (t)x + a 2 (t)x 2 +... + a n (t)x n, ẋ 2 =... a 2 (t)x + a 22 (t)x 2 +... + a 2n (t)x n, (5) ẋ n = a n (t)x + a n2 (t)x 2 +... + a nn (t)x n àáî ó ìàòðè íié ôîðìi: äå Ẋ(t) = A(t)X(t), (6) A(t) = a (t) a 2 (t)... a n (t) a 2 (t) a 22 (t)... a 2n (t)............ a n (t) a n2 (t)... a nn (t), X(t) = x (t) x 2 (t)... x n (t)  îáëàñòi íåïåðåðâíîñòi êîåôiöi¹íòiâ a ij (t), i, j =, n, ñèñòåìà (5) çàäîâîëüíÿ¹ óìîâè òåîðåìè iñíóâàííÿ òà ¹äèíîñòi ðîçâ'ÿçêó çàäà i Êîøi. Ôóíäàìåíòàëüíîþ ñèñòåìîþ ðîçâ'ÿçêiâ ñèñòåìè (5) íàçèâàþòü ñóêóïíiñòü äîâiëüíèõ n ëiíiéíî íåçàëåæíèõ ðîçâ'ÿçêiâ X k (t) = (x (k) (t),x(k) 2 (t),...,x(k) n (t)), k =, n. ßêùî X k (t), k =, n, ôóíäàìåíòàëüíà ñèñòåìà ðîçâ'ÿçêiâ ñèñòåìè (5), òî çàãàëüíèé ðîçâ'ÿçîê ñèñòåìè ì๠âèãëÿä X(t) = n C k X k (t), äå C, C 2,..., C n äîâiëüíi ñòàëi. Ó âèïàäêó, êîëè ìàòðèöÿ A(t) ó ïðàâié àñòèíi (6) íå çàëåæèòü âiä t, äëÿ âiäøóêàííÿ ôóíäàìåíòàëüíî ñèñòåìè ðîçâ'ÿçêiâ âèêîðèñòîâóþòü ìåòîäè ëiíiéíî àëãåáðè. Äëÿ öüîãî çíàõîäÿòü âëàñíi èñëà λ, λ 2,..., λ s ìàòðèöi A(t) = a a 2... a n a 2 a 22... a 2n............ a n a n2... a nn 2 k=
ÿê êîðåíi õàðàêòåðèñòè íîãî ðiâíÿííÿ det(a λe) =. (7) Äëÿ êîæíîãî êîðåíÿ (ç óðàõóâàííÿì éîãî êðàòíîñòi) âèçíà àþòü âiäïîâiäíèé éîìó àñòèííèé ðîçâ'ÿçîê X (λ k) (t). Çàãàëüíèé ðîçâ'ÿçîê ñèñòåìè ì๠âèãëÿä X(t) = s C k X (λk) (t). (8) k= Ïðè öüîìó ìîæëèâi òàêi âèïàäêè: a) ÿêùî λ - äiéñíèé êîðiíü êðàòíîñòi, òî X (λ) (t) = Y (λ) e λt = y (λ) y (λ) 2... y (λ) n eλt, (9) äå Y (λ) - âëàñíèé âåêòîð ìàòðèöi A, ùî âiäïîâiä๠âëàñíîìó çíà åííþ λ (íàãàäà¹ìî, ùî AY (λ) = λy (λ), Y (λ) ); b) ÿêùî λ - óÿâíèé êîðiíü êðàòíîñòi, òî êîðåíåì õàðàêòåðèñòè íîãî ðiâíÿííÿ (7) ¹ òàêîæ ñïðÿæåíå äî λ èñëî λ. ßêùî ìàòðèöÿ A ñêëàäà¹òüñÿ ç äiéñíèõ èñåë, òî çàìiñòü óÿâíèõ àñòèííèõ ðîçâ'ÿçêiâ X (λ) (t) òà X ( λ) (t) äîñòàòíüî âçÿòè äiéñíi àñòèííi ðîçâ'ÿçêi ó âèãëÿäi X (λ) (t) = ReX (λ) (t) òà X (λ) 2 (t) = ImX (λ) (t) ; c) ÿêùî λ - êîðiíü êðàòíîñòi r 2, òî âiäïîâiäíèé öüîìó êîðåíþ ðîçâ'ÿçîê ñèñòåìè (6) çíàõîäÿòü ó âèãëÿäi âåêòîðà X (λ) (t) = α () + α (2) 2 + α (2) 2......... n + α n (2) α () α () 3 t +... + α(r) tr t +... + α(r) 2 tr t +... + α n (r) t r eλt, ()
äå êîåôiöi¹íòè α (j) i, i =, 2,..., n; j =, 2,..., r, âèçíà àþòü iç ñèñòåìè ëiíiéíèõ ðiâíÿíü, ÿêi äiñòàþòü ïîðiâíÿííÿì êîåôiöi¹íòiâ ïðè îäíàêîâèõ ñòåïåíÿõ t ïiñëÿ ïiäñòàíîâêè âåêòîðà () ó ñèñòåìó (6). Íîðìàëüíà ëiíiéíà íåîäíîðiäíà ñèñòåìà äèôåðåíöiàëüíèõ ðiâíÿíü ó ìàòðè íié ôîðìi ì๠âèãëÿä Ẋ(t) = A(t)X(t) + F (t), () äå F (t) = (f (t), f 2 (t),..., f n (t)). ßêùî âiäîìî äåÿêèé àñòèííèé ðîçâ'ÿçîê X(t) ñèñòåìè (), òî çàãàëüíèé ðîçâ'ÿçîê ì๠âèãëÿä: X(t) = X (t) + X(t), (2) äå X (t) - çàãàëüíèé ðîçâ'ÿçîê âiäïîâiäíî () îäíîðiäíî ñèñòåìè ðiâíÿíü Ẋ(t) = A(t)X(t). ßêùî âiäîìî ôóíäàìåíòàëüíó ñèñòåìó X k (t), k =, 2,..., n, ðîçâ'ÿçêiâ âiäïîâiäíî îäíîðiäíî ñèñòåìè, òî çàãàëüíèé ðîçâ'ÿçîê íåîäíîðiäíî ñèñòåìè çàâæäè ìîæíà çíàéòè ìåòîäîì âàðiàöi äîâiëüíèõ ñòàëèõ. À ñàìå, ïîêëàäàþ è X(t) = n C k (t)x k (t), (3) k= âèçíà à¹ìî ôóíêöi C k (t), ïiäñòàâëÿþ è (3) ó ñèñòåìó (). Âðàõîâóþ è ïðè öüîìó ðiâíiñòü Ẋ k (t) A(t)X k (t) =, k =, 2,..., n, äiñòà¹ìî ñèñòåìó ðiâíÿíü âiäíîñíî Ċk(t): n Ċ k (t)x k (t) = F (t). (4) k= Ç öi¹ ñèñòåìè çíàõîäèìî ôóíêöi Ċ k (t) = ϕ k (t), iíòåãðóþ è ÿêi, âèçíà à¹ìî C k (t) ç òî íiñòþ äî ñòàëèõ. Ïiäñòàâëÿþ è õ ó (3), îòðèìó¹ìî øóêàíèé çàãàëüíèé ðîçâ'ÿçîê íåîäíîðiäíî ñèñòåìè (). 4
ßêùî êîåôiöi¹íòè a ij (t) ñèñòåìè () ñòàëi, òîáòî a ij (t) = a ij, i, j =, 2,..., n, à ôóíêöi f i (t) ìàþòü ñïåöiàëüíèé âèãëÿä (P (t) cos βt + Q(t) sin βt)e αt, (5) äå P (t) òà Q(t) - ìíîãî ëåíè, àñòèííèé ðîçâ'ÿçîê X(t) çíàõîäÿòü ìåòîäîì íåâèçíà åíèõ êîåôiöi¹íòiâ. ßêùî α λ i, i =, s, òî X(t) øóêàþòü ó âèãëÿäi, àíàëîãi íîìó (5), òîáòî X(t) = (F (t) cos βt + F 2 (t) sin βt)e αt, äå F (t)if 2 (t) ìíîãî ëåíè ñòåïåíÿ k ( k ìàêñèìàëüíèé ñåðåä ñòåïåíiâ ìíîãî ëåíiâ P (t) òà Q(t)). ßêùî çíàéäåòüñÿ òàêe âëàñíå çíà åííÿ λ i = α+iβ êðàòíîñòi r, ùî α + iβ = λ i, òî X(t) øóêàþòü ç óðàõóâàííÿì çáiæíîñòi êëþ îâîãî èñëà α ± iβ ç êîðåíÿìè õàðàêòåðèñòè íîãî ðiâíÿííÿ. ßêùî k - íàéáiëüøèé ñòåïiíü ìíîãî ëåíiâ P (t) òà Q(t) i λ i = α+iβ êîðiíü êðàòíîñòi r õàðàêòåðèñòè íîãî ðiâíÿííÿ, òî àñòèííèé ðîçâ'ÿçîê X(t) øóêàþòü ó âèãëÿäi X(t) = Re tr γ t k+ + γ t k +... + γ,k+ γ 2 t k+ + γ 2 t k +... + γ 2,k+... γ n t k+ + γ n t k +... + γ n,k+ eλt (6) Ó áàãàòüîõ âèïàäêàõ çàäà ó Êîøi çðó íî ðîçâ'ÿçóâàòè çà äîïîìîãîþ ìàòðè íî åêñïîíåíòè. Åêñïîíåíòîþ ìàòðèöi e A íàçèâàþòü ñóìó ðÿäó e A := I +! A + 2! A2 +... + n! An +... = k= k! Ak, (7) äå I îäèíè íà ìàòðèöÿ. Îñêiëüêè ìàòðèöÿ X(t) = e At ¹ ðîçâ'ÿçêîì ìàòðè íî çàäà i Êîøi X = AX, X() = I (6), òî çàäà à iíòåãðóâàííÿ öi¹ ñèñòåìè çâîäèòüñÿ äî çíàõîäæåííÿ åêñïîíåíòè âiäïîâiäíî ìàòðèöi. Ìàòðè íó åêñïîíåíòó çðó íî áóäóâàòè çâåäåííÿì ìàòðèöi äî æîðäàíîâî ôîðìè J(A). Âiäîìî, ùî iñíóþòü òàêi ìàòðèöi T, ùî A = T J(A)T. Íàãàäà¹ìî, ùî J(A) = 5
diag(j m (λ ), J m2 (λ 2 ),..., J ms (λ s )), äå J mi (λ i ) æîðäàíîâà êëiòèíà, òîáòî J mi (λ i ) = λ i... λ i........... λ i m i ðîçìið æîðäàíîâî êëiòèíè, s i=, m i = n ïîðÿäîê ñèñòåìè. Òàêèì èíîì, e At = T e J(A)t T, äå e J(A)t = diag(e J m (λ )t, e J m 2 (λ 2 )t,...,e J ms(λ s )t ). Îñêiëüêè J mi (λ i )t = λ i + E, äå E =.............., òî e Jm i (λ i)t = e λ it e E Ìàòðèöþ e Et íåâàæêî ïîáóäóâàòè çà äîïîìîãîþ ðÿäó (7), îñêiëüêè E m i =. Ïðèêëàä. Ðîçâ'ÿçàòè çàäà ó Êîøi x = x +y z, y = x +2y z, z = 2x y +4z,, X() = Çíàéäåìî çàãàëüíié ðîçâ'ÿçîê ñèñòåìè. Äëÿ öüîãî îá èñëèìî âëàñíi çíà åííÿ òà âëàñíi âåêòîðè âiäïîâiäíî ìàòðèöi. Âëàñíi çíà åííÿ ìàòðèöi A çíàéäåìî ç õàðàêòåðèñòè íîãî ðiâíÿííÿ λ 2 λ 2 4 λ Ìà¹ìî λ = λ 2 = 2, λ 3 = 3. =, (λ 2)2 (λ 3) =. 6
Çíàéäåìî âiäïîâiäíi âëàñíi âåêòîðè. Êîîðäèíàòè âëàñíîãî âåêòîðà, âiäïîâiäíîãî âëàñíîìó çíà åííþ λ 3 = 3, âèçíà èìî iç ñèñòåìè ðiâíÿíü 2x +y z =, x y z =, 2x y +z =. Îäíèì iç ðîçâ'ÿçêiâ ðiâíÿííÿ ¹ x = 2, z = 3, y =. Âëàñíèé âåêòîð ì๠âèãëÿä ϕ λ 3 = {2, 3, }. Îòæå, îòðèìàëè îäèí iç ðîçâ'ÿçêiâ ôóíäàìåíòàëüíî ñèñòåìè ëiíiéíî íåçàëåæíèõ ðîçâ'ÿçêiâ X λ 3 = ϕ λ3 e 2 Äâà iíøèõ ðîçâ'ÿçêè øóêàòèìåìî ó âèãëÿäi X λ,2 = a t + b a 2 t + b 2 a 3 t + b 3 e 2 Çíàéäåìî êîíñòàíòè a i, b i, i =, 2, 3 ìåòîäîì íåâèçíà åíèõ êîåôiöi¹íòiâ. Äëÿ öüîãî ïiäñòàâèìî ðîçâ'ÿçîê X λ,2 ó ñèñòåìó a e 2t +2(a t + b )e 2t = (a t + b )e 2t + (a 2 t + b 2 )e 2t (a 3 t + b 3 )e 2t, a 2 e 2t +2(a 2 t + b 2 )e 2t = (a t + b )e 2t + 2(a 2 t + b 2 )e 2t (a 3 t + b 3 )e 2t, a 3 e 2t +2(a 3 t + b 3 )e 2t = 2(a t + b )e 2t (a 2 t + b 2 )e 2t + 4(a 3 t + b 3 )e 2 Ïîðiâíþþ è êîåôiöi¹íòè ïðè ëiíiéíî íåçàëåæíèõ êîìïîíåíòàõ e 2t i te 2t êîæíîãî ðiâíÿííÿ äàíî ñèñòåìè, ìà¹ìî a + 2b = b + b 2 b 3, 2a = a + a 2 a 3, a 2 + 2b 2 = b + 2b 2 b 3, 2a 2 = a + 2a 2 a 3, a 3 + 2b 3 = 2b b 2 + 4b 3, 2a 3 = 2a a 2 + 4a 3. 7
Ïiñëÿ øòó íèõ ïåðåòâîðåíü äiñòà¹ìî ñïðîùåíó ñèñòåìó a = a 3, a 2 =, b + b 2 b 3 = a, b b 3 =, 2b b 2 + 2b 3 = a 3. Îñêiëüêè öÿ ñèñòåìà ì๠áåçëi ðîçâ'ÿçêiâ, çíàéäåìî äåÿêèé áàçèñíèé ðîçâ'ÿçîê. Äëÿ öüîãî ïîêëàäåìî C = a, C 2 = b, òîäi a 3 = C, b 2 = C, b 3 = C 2. Çàãàëüíèé ðîçâ'ÿçîê ñèñòåìè ì๠âèãëÿä X = C t + C 2 C C t C 2 àáî ó ñêàëÿðíié ôîðìi: e 2t + C 3 2 3 e 3t x = (C t + C 2 )e 2t + 2C 3 e 3t, y = C e 2t + C 3 e 3t, z = (C t + C 2 )e 2t 3C 3 e 3 Çíàéäåìî àñòèííèé ðîçâ'ÿçîê, ùî âiäïîâiä๠ïî àòêîâèì óìîâàì. Äëÿ öüîãî ïiäñòàâèìî ïî àòêîâi äàíi ó çàãàëüíèé ðîçâ'ÿçîê. Äiñòàíåìî: = C 2 + 2C 3, = C + C 3, = C 2 3C 3, i ðîçâ'ÿæåìî ñèñòåìó âiäíîñíî íåâiäîìèõ C i, i =, 2, 3. Ðîçâ'ÿçêîì ¹ C 3 =, C 2 =, C =. Òîäi ðîçâ'ÿçîê çàäà i Êîøi ì๠âèãëÿä X(t) = te 2t e 2t te 2t Ðîçâ'ÿæåìî öþ çàäà ó çà äîïîìîãîþ ìàòðèöàíòà e A Ó íàøîìó âèïàäêó A = 2 2 4 8
Çíàéäåìî âëàñíi çíà åííÿ ìàòðèöi A ç õàðàêòåðèñòè íîãî ðiâíÿííÿ λ 2 λ 2 4 λ =, (λ 2)2 (λ 3) =. Ìà¹ìî λ = λ 2 = 2, λ 3 = 3. Æîðäàíîâà ôîðìà ìàòðèöi A ì๠âèãëÿä J = 2 2 3 Ìàòðèöþ T = (a ij ), òàêó ùî A = T JT, çíàõîäèìî ç ìàòðè íîãî ðiâíÿííÿ a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 2 2 4 = 2 2 3 = a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 Ïåðåìíîæàþ è ìàòðèöi ó ëiâié i ïðàâié àñòèíàõ îñòàííüî ðiâíîñòi, äiñòà¹ìî: a a 2 + 2a 3 a + 2a 2 a 3 a a 2 + 4a 3 a 2 a 22 + 2a 23 a 2 + 2a 22 a 23 a 2 a 22 + 4a 23 a 3 a 32 + 2a 33 a 3 + 2a 32 a 33 a 3 a 32 + 4a 33 = = 2a + a 2 2a 2 + a 22 2a 3 + a 23 2a 2 2a 22 2a 23 3a 3 3a 32 3a 33 9
Ìà¹ìî ñèñòåìó ç 9 ðiâíÿíü ç 9 íåâiäîìèìè: a a 2 + 2a 3 = 2a + a 2, a + 2a 2 a 3 = 2a 2 + a 22, a a 2 + 4a 3 = 2a 3 + a 23, a 2 a 22 + 2a 23 = 2a 2, a 2 + 2a 22 a 23 = 2a 22, a 2 a 22 + 4a 23 = 2a 23, a 3 a 32 + 2a 33 = 3a 3, a 3 + 2a 32 a 33 = 3a 32, a 3 a 32 + 4a 33 = 3a 33. Øòó íèìè ïiäñòàíîâêàìè ðîçâ'ÿçó¹ìî äàíó ñèñòåìó i çíàõîäèìî ìàòðèöi T = 3 2, T = Çíàéäåìî e J Îñêiëüêè J = D + E, äå D = 2 2 3, E = 2 3 i ìàòðèöi D òà E êîìóòóþòü, òîáòî ([D, E] =, DE ED = ), òî e Jt = e (D+E)t = e Dt e Et i e Dt = e 2t e 2t e 3t Ìàòðèöþ e Et çíàéäåìî çà äîïîìîãîþ ðÿäó (7). Îñêiëüêè E 2 =, òî e Et = I +! E = + t = t,
Òîäi e Jt = e At = = e 2t e 2t e 3 2 3 t e 2t te 2t e 2t e 3t = e Jt = e 2t te 2t e 2t e 3t 3 2 e 2t (3 + t) 2e 3t te 2t e 2t (2 + t) 2e 3t e 2t e 3t e 2t e 2t e 3t (3 + t)e 2t + 3e 3t te 2t e 2t (2 + t) + 3e 3t Îòæå, ðîçâ'ÿçîê çàäàíî çàäà i Êîøi ì๠âèãëÿä x(t) = e At = te 2t e 2t te 2t Ïðèêëàä 2. Çíàéòè çàãàëüíèé ðîçâ'ÿçîê ñèñòåìè x = 3x + 2(y + z) 2te t + e t 2e 2t + 2, y = 2(x + y + z) 2te t + 2, z = 3(x + y) 2z + 3te t + 3e 2 = Çíàéäåìî çàãàëüíèé ðîçâ'ÿçîê âiäïîâiäíî îäíîðiäíî ñèñòåìè ðiâíÿíü x = 3x + 2y + 2z, y = 2x + 2y + 2z, z = 3x 3y 2z. Âèçíà èìî âëàñíi çíà åííÿ òà âëàñíi âåêòîðè âiäïîâiäíî ìàòðèöi. Âëàñíi çíà åííÿ ìàòðèöi A çíàéäåìî ç õàðàêòåðèñòè íîãî ðiâíÿííÿ 3 λ 2 2 2 2 λ 2 3 3 2 λ =, (λ )((λ )2 + ) =.
Ìà¹ìî λ =, λ 2 = + i, λ 3 = i. Çíàéäåìî âiäïîâiäíi âëàñíi âåêòîðè. Êîîðäèíàòè âëàñíîãî âåêòîðà, ùî âiäïîâiäàþòü âëàñíîìó çíà åííþ λ =, âèçíà èìî iç ñèñòåìè ðiâíÿíü 2x +2y +2z =, 3x +y +2z =, 3x 3y 3z =. Îäíèì iç ðîçâ'ÿçêiâ äàíî ñèñòåìè ¹ x =, y =, z =. Âëàñíèé âåêòîð ì๠âèãëÿä ϕ λ = {,, }. Òàêèì èíîì, ìè îòðèìàëè îäèí iç ðîçâ'ÿçêiâ ϕ λ e t ôóíäàìåíòàëüíî ñèñòåìè ëiíiéíî íåçàëåæíèõ ðîçâ'ÿçêiâ. Çíàéäåìî âëàñíèé âåêòîð, ùî âiäïîâiä๠âëàñíîìó çíà åííþ λ 2 = + i. Äëÿ öüîãî ðîçâ'ÿæåìî ñèñòåìó (2 i)x +2y +2z =, 3x +( i)y +2z =, 3x 3y (3 + i)z =. Îäíèì iç ðîçâ'çêiâ ñèñòåìè ¹ x = 2 2i, y = 2, z = 3 + 3i. Âëàñíèé âåêòîð ì๠âèãëÿä ϕ λ = {2 2i, 2, 3 + 3i}. Äâà iíøi ëiíiéíî íåçàëåæíi ðîçâ'ÿçêè ôóíäàìåíòàëüíî ñèñòåìè âåêòîðiâ çíàéäåìî ó âèãëÿäi Reϕ λ e(+i)t òà Imϕ λ e(+i) Îñêiëüêè ϕ λ e(+i)t = ϕ λ et (cos t + i sin t) = = 2 2i, 2, 3 + 3i e t (cos t + i sin t) = òî = 2(cos t + sin t) 2i(cos t sin t), 2 cos t +2i sin t, 3(cos t + sin t) +3i(cos t sin t) Reϕ λ e(+i)t = 2(cos t + sin t) 2 cos t 3(cos t + sin t) 2 e t e t,
òà Imϕ λ e(+i)t = 2(cos t + sin t), 2 sin t, 3(cos t sin t) e Îòæå, çàãàëüíèé ðîçâ'ÿçîê îäíîðiäíî ñèñòåìè ðiâíÿíü ìàòèìå âèãëÿä X = C e t àáî ó ñêàëÿðíié ôîðìi: + C 2 +C 3 2(cos t + sin t) 2 cos t 3(cos t + sin t) 2(cos t + sin t), 2 sin t, 3(cos t sin t) e t + e t x = C e t +2C 2 (cos t + sin t)e t 2C 3 (cos t + sin t)e t, y = +2C 2 cos te t +2C 3 sin te t, z = C e t 3C 2 (cos t + sin t)e t +3C 3 (cos t sin t)e Ïðàâà àñòèíà ñèñòåìè, çàäàíî â óìîâi, ì๠ñïåöiàëüíèé âèãëÿä, òîìó àñòèííèé ðîçâ'ÿçîê øóêàòèìåìî, âèêîðèñòîâóþ- è ìåòîä íåâèçíà åíèõ êîåôiöi¹íòiâ. Öåé ñïîñiá ¹ äîöiëüíiøèì, îñêiëüêè àëãåáðà íi äi ¹ ïðîñòiøèìè, íiæ ïðîöåäóðà iíòåãðóâàííÿ. Çíàéäåìî àñòèííèé ðîçâ'ÿçîê îäíîðiäíî ñèñòåìè ðiâíÿííÿ ó âèãëÿäi ñóïåðïîçèöi ðîçâ'ÿçêiâ X = X + X 2 + X 3, äå X, X 2 i X 3 âiäïîâiäàþòü te t, e 2t òà const ó ïðàâié àñòèíi çàäàíî ñèñòåìè. Êëþ îâèìè èñëàìè ó íàøîìó âèïàäêó ¹ ñòåïåíi åêñïîíåíòè, òîáòî,, 2. Îñêiëüêè âëàñíå çíà åííÿ (ÿêå ì๠êðàòíiñòü ) õàðàêòåðèñòè íî ìàòðèöi çáiãà¹òüñÿ ç êëþ îâèì èñëîì (ñòåïåíÿ åêñïîíåíòè), òî X øóêàòèìåìî ó âèãëÿäi: X = A t 2 + B t + D A 2 t 2 + B 2 t + D 2 A 3 t 2 + B 3 t + D 3 e t, 3
äå áåðåìî ìíîãî ëåí ñòåïåíÿ íà îäèíèöþ áiëüøå, íiæ ñòåïiíü ìíîãî ëåíà ïðè e t ó ðiâíÿííi, i äîìíîæà¹ìî íà t ó ñòåïåíi íà îäèíèöþ ìåíøå, íiæ êðàòíiñòü âiäïîâiäíîãî âëàñíîãî çíà åííÿ. Äâi iíøi êîìïîíåíòè àñòèííîãî ðîçâ'ÿçêó çíàõîäèìî ó âèãëÿäi X 2 = E E 2 E 3, e 2t, X3 = Ïiäñòàâèìî ó çàäàíó ñèñòåìó çíà åííÿ X, âèêëþ àþ è âiëüíi ëåíè òà ëåíè, ùî ìiñòÿòü e 2 Ìà¹ìî: F F 2 F 3 (A t 2 +B t + D )e t + (2A t + B )e t = = 3(A t 2 + B t + D )e t + 2(A 2 t 2 + B 2 t + D 2 )e t + + 2(A 3 t 2 + B 3 t + D 3 )e t 2te t + e t, (A 2 t 2 +B 2 t + D 2 )e t + (2A 2 t + B 2 )e t = = 2(A t 2 + B t + D )e t + 2(A 2 t 2 + B 2 t + D 2 )e t + + 2(A 3 t 2 + B 3 t + D 3 )e t 2te t, (A 3 t 2 +B 3 t + D 3 )e t + (2A 3 t + B 3 )e t = = 3(A t 2 + B t + D )e t 3(A 2 t 2 + B 2 t + D 2 )e t 2(A 3 t 2 + B 3 t + D 3 )e t + 3te Ïîðiâíÿ¹ìî êîåôiöi¹íòè ïðè ëiíiéíî íåçàëåæíèõ ëåíàõ te t, e t òà êîíñòàíòàõ i ñêëàäåìî ñèñòåìó: A = 3A + 2A 2 + 2A 3, B + 2A = 3B + 2B 2 + 2B 3 2, D + B = 3D + 2D 2 + 2D 3 +, A 2 = 2A + 2A 2 + 2A 3, B 2 + 2A 2 = 2B + 2B 2 + 2B 3 2, D 2 + B 2 = 2D + 2D 2 + 2D 3, A 3 = 3A 3A 2 2A 3, B 3 + 2A 3 = 3B 3B 2 32B 3 + 3, D 3 + B 3 = 3D 3D 2 32D 3. Ðîçâ'ÿçóþ è îñòàííþ ñèñòåìó øòó íèìè ïiäñòàíîâêàìè, çíàõîäèìî A i = D i =, i =, 2, 3; B 2 = B 3 =, B =. Ôðàãìåíò àñòèííîãî ðîçâ'ÿçêó ì๠âèãëÿä X = 4 te t.
Ïiäñòàâèìî ó çàäàíó ñèñòåìó çíà åííÿ X 2, âèêëþ àþ è âiëüíi ëåíè òà ëåíè, ùî ìiñòÿòü e Äiñòà¹ìî: (2E )e 2t = (3E + 2E 2 + 2E 3 )e 2t 2e 2t, (2E 2 )e 2t = (2E + 2E 2 + 2E 3 )e 2t, (2E 3 )e 2t = ( 3E 2E 2 2E 3 )e 2t + 3e 2t àáî 2 = E + 2E 2 + 2E 3, = 2E + 2E 3, 3 = 3E + 3E 2 + 3E 3. Ðîçâ'ÿçóþ è äàíó ñèñòåìó øòó íèìè ïiäñòàíîâêàìè, çíàõîäèìî E = E 3 =, E 2 =. Ôðàãìåíò àñòèííîãî ðîçâ'ÿçêó ì๠âèãëÿä X 2 = e 2t. Ïiäñòàâèìî ó çàäàíó ñèñòåìó çíà åííÿ X 3, âèêëþ àþ è ëåíè, ùî ìiñòÿòü e t òà e 2 Ìà¹ìî: = 3F + 2F 2 + 2F 3 + 2, = 2F + 2F 2 + 2F 3 + 2, = 3F 3F 2 2E 3. Ðîçâ'çóþ è äàíó ñèñòåìó øòó íèìè ïiäñòàíîâêàìè, ïîìi à¹ìî, ùî F = F 2 =, F 3 =. Ôðàãìåíò àñòèííîãî ðîçâ'ÿçêó ì๠âèãëÿä X 3 =. àñòèííèé ðîçâ'ÿçîê ñèñòåìè X = Òîäi çàãàëüíèé ðîçâ'ÿçîê çàïèøåòüñÿ ó âèãëÿäi X =C e t + C 3 + C 2 2(cos t + sin t), 2 sin t, 3(cos t sin t) 2(cos t + sin t) 2 cos t 3(cos t + sin t) 5 e t + te t e 2t e t + te t e 2t.
àáî ó ñêàëÿðíié ôîðìi: x = C e t + 2C 2 (cos t + sin t)e t 2C 3 (cos t + sin t)e t + te t, y = 2C 2 cos te t + 2C 3 sin te t + e 2t, z = C e t 3C 2 (cos t + sin t)e t + 3C 3 (cos t sin t)e t. Ïðèêëàä 3. Çíàéòè çàãàëüíèé ðîçâ'ÿçîê ñèñòåìè x = x + 2y + 3z + te t, y = y + 2z, Çíàéäåìî çàãàëüíèé ðîçâ'ÿçîê âiäïîâiäíî îäíîðiäíî ñèñòåìè ðiâíÿíü x = x + 2y + 3z, y = y + 2z, z = z. Îá èñëèìî âëàñíi çíà åííÿ òà âëàñíi âåêòîðè âiäïîâiäíî ìàòðèöi. Âëàñíi çíà åííÿ ìàòðèöi A çíàéäåìî ç õàðàêòåðèñòè íîãî ðiâíÿííÿ λ 2 3 λ 2 λ =, (λ )3 =. Ìà¹ìî λ = êðàòíîñòi 3. Ðîçâ'ÿçîê îäíîðiäíî ñèñòåìè ðiâíÿíü øóêàòèìåìî ó âèãëÿäi X (λ) = a t 2 + b t + d a 2 t 2 + b 2 t + d 2 a 3 t 2 + b 3 t + d 3 e Ïiäñòàâèìî çíà åííÿ X (λ) ó çàäàíó ñèñòåìó. Äiñòàíåìî: (a t 2 +b t + d )e t + (2a t + b )e t = (a t 2 + b t + d )e t + 2(a 2 t 2 + b 2 t + d 2 )e t + 3(a 3 t 2 + b 3 t + d 3 )e t, (a 2 t 2 +b 2 t + d 2 )e t + (2a 2 t + b 2 )e t = (a 2 t 2 + b 2 t + d 2 )e t + 2(a 3 t 2 + b 3 t + d 3 )e t, (a 3 t 2 +b 3 t + d 3 )e t + (2a 3 t + b 3 )e t = (a 3 t 2 + b 3 t + d 3 )e 6
Ïîðiâíÿ¹ìî êîåôiöi¹íòè ïðè ëiíiéíî íåçàëåæíèõ ëåíàõ t 2 e t, te t òà e Ìà¹ìî: a = a + 2a 2 + 3a 3, b + 2a = b + 2b 2 + 3b 3, d + b = d + d 2 + d 3, a 2 = a 2 + 2a 3, b 2 + 2d 2 = b 2 + 2b 3, d 2 + b 2 = d 2 + 2d 3, a 3 = a 3, b 3 + 2a 3 = b 3, d 3 + b 3 = d 3. Äàíà ñèñòåìà ì๠áåçëi ðîçâ'ÿçêiâ. Âèáåðåìî äåÿêèé áàçèñíèé ðîçâ'ÿçîê, ïîêëàäàþ è a = C, d = C 2 òà d 2 = C 3, çîêðåìà a 2 = a 3 = b 3 =. Òîäi b = 5 2 C, b 2 = C, d 3 = 2 C. Ðîçâ'ÿçîê îäíîðiäíî ñèñòåìè, âiäïîâiäíî çàäàíié, çàïèøåòüñÿ ó âèãëÿäi: x = (C t 2 + 5 2 C t + C 2 )e t, y = (C t + C 3 )e t, z = 2 C e Ðîçâ'ÿçîê íåîäíîðiäíî ñèñòåìè,âiäïîâiäíî çàäàíié, çíàéäåìî ìåòîäîì âàðiàöi äîâiëüíèõ ñòàëèõ C, C 2 òà C 3. Äëÿ öüîãî ñêëàäåìî ñèñòåìó (4) (C t2 + 5 2 C t + C 2 )et = te t, (C t + C 3 )et =, 2 C et = et Ðîçâ'ÿæåìî äàíó ñèñòåìó âiäíîñíî íåâiäîìèõ C, C 2 òà C 3. Ìà¹ìî: C = 2 t, C 2 = t + 5, C 3 = 2. Ïðîiíòåãðóâàâøè, çíàéäåìî: C = 2 ln t + C, C 2 = 2 t2 + 5t + C 2, C 3 = 2t + C 3. 7
Çàãàëüíèé ðîçâ'ÿçîê íåîäíîðiäíî ñèñòåìè ìàòèìå âèãëÿä x = ( C t 2 + 5 2 C t + C 2 )e t + (2t 2 ln t + 5 2 t ln t 2 t2 + 5t)e t, y = ( C t + C 3 )e t + (2t ln t 2t)e t, z = 2 C e t + e t ln t. Çàâäàííÿ äëÿ ñàìîñòiéíîãî âèêîíàííÿ I. Äëÿ çàäàíî ñèñòåìè äèôåðåíöiàëüíèõ ðiâíÿíü çíàéòè çàãàëüíèé ðîçâ'ÿçîê ìàòðè íèì ìåòîäîì òà ðîçâ'ÿçàòè âêàçàíó çàäà ó Êîøi. II. Îá èñëèòè ìàòðèöàíò âiäïîâiäíî ìàòðèöi çàäàíî ñèñòåìè äèôåðåíöiàëüíèõ ðiâíÿíü òà çà éîãî äîïîìîãîþ ðîçâ'ÿçàòè âêàçàíó çàäà ó Êîøi; ïîðiâíÿòè îòðèìàíèé ðîçâ'ÿçîê iç ðîçâ'ÿçêîì ïîïåðåäíüîãî çàâäàííÿ.. 2. 3. 4. x = 7x +y 5z, y = 3x 2y 3z, z = 8x y +6z, x = 4x +y 3z, y = 2x y 2z, z = 5x y +4z, x = 6x +y +5z, y = 2x +y +2z, z = 7x y 6z, x = 5x +y +3z, y = x +2y +z, z = 4x y 2z, X() = X() = X() = X() = 5. x = 2x +y +9z, y = 4x +3y +4z, z = 3x y z, 8 X() =
6. x = 3x +y z, y = x 2y z, z = 2x y, X() = 7. 8. x = 2x +y +3z, y = x y +z, z = 4x y 5z, x = 8x +y +7z, y = 3x +y +3z, z = x y 9z, X() = X() = 9. x = x +y +9z, y = 4x +2y +4z, z = 3x y z, X() =. x = 4x +y +z, y = 5x +3y +5z, z = 6x y 3z, X() =. 2. x = 9x +y 7z, y = 4x 2y 4z, z = x y +9z, x = 6x +y 5z, y = 3x y 3z, z = 8x y +7z, X() = X() = 3. x = y z, y = y x z, z = 2x y +3z, X() = 9
4. x = y +5x +3z, y = z +x +2y, z = 4x y 2z, X() = 5. x = y +8x +5z, y = 2x +2z +3y, z = 7x y 4z, X() = 6. x = x +y +3z, y = x +z 2y, z = 4x y 6z, X() = 7. x = 4x +y +5z, y = 2x y +2z, z = 7x y 8z, X() = 8. x = x +9z +y, y = 4x +y +4z, z = 3x y 2z, X() = 9. x = 3x +z +y, y = 2y +5x +5z, z = 6x 4z y, X() = 2 2. x = 6x +3z +y, y = 3y +6z +6x, z = 9x y 6z, X() = III. Çíàéòè çàãàëüíèé ðîçâ'ÿçîê ñèñòåìè äèôåðåíöiàëüíèõ ðiâíÿíü, âèêîðèñòîâóþ è ìåòîä íåâèçíà åíèõ êîåôiöi¹íòiâ äëÿ îá èñëåííÿ àñòèííîãî ðîçâ'ÿçêó íåîäíîðiäíî ñèñòåìè.. x = 2(y x) + e 2t 2e t, y = 2(x + z) 2te 2t e t, z = 3y 2z + 3e t + 2. 2
2. x = x + 4y + 2z + e t 2te t 4e t 4, y = 2y + 4(x + z) 4te t e t 8, z = (3x + 6y + 4z) + 3te t + 6e t + 8. 3. 4. 5. 6. 7. 8. 9. x = 3x + 2(y + z) 2te t + e t 2e t 6, y = 2(x + y + z) 2te t 3e t 6, z = 3(x + y) 2z + 3te t + 3e t + 6. x = 2x + 4y + z te 2t + e 2t 4e t 4, y = y + 4(x + z) 4te 2t 2e t 6, z = (5x + 6y + 3z) + 5te 2t + 6e t + 52. x = 4x + 2(y z) + 2te 2t + e 2t 2e t +, y = y + 2(x + z) 2te 2t, z = 3(x y) + z 3te 2t + 3e t 5. x = x + 4y 4e t + e t, y = 4(x + z) + 3y 4te t 2e t 24, z = (z + 6y) + 6e t + 2. x = 7x + 2y + 6z 6te t + e t 7e t 42, y = 2(x + z) 2te t e t 4, z = 3(3x + y) 8z + 9te t + 3e t + 56. x = 2(5x + 2(y + 2z)) 8te 2t + e 2t 4e t 64, y = 4(x + z) + 2y 4te 2t e t 32, z = 6(2x + y) z + 2te 2t + 6e t + 8. x = 2(y 3x 2z) + 4te 2t + e 2t 2e t + 36, y = 2(x + y + z) 2te 2t e t 8, z = 6x 3y + 4z 6te 2t + 3e t 36. 2
. x = 3x + 4(y + z) 4te t + e t 4e t 4, y = y + 4(x + z) 4te t 4, z = 6(x + y) 7z + 6te t + 6e t + 7.. 2. 3. 4. 5. 6. 7. x = 5x + 2y + 4z 4te t + e t 2e t 44, y = 2(x + z) + y 2te t 2e t 22, z = 6x 3y 5z + 6te t + 3e t + 55. x = 2(3z + 2y + 4x) 6te 2t 4e t + e 2t 72, y = 4(x + z) + 3y 4te 2t 2e t 48, z = 3(3x + 2y) 7z + 9te 2t + 6e t + 84. x = 2(y 4x 3z) + 6te 2t 2e 2t + e 2t + 78, y = 2(x + z) + 3y 2te 2t e 2t 26, z = 3(3x y) + 7z 9te 2t + 3e 2t 9. x = 2y x + e t 2e t, y = y + 2(x + z) 2te t e t 28, z = (z + 3y) + 3e t + 4. x = 5x + 4(y + z) 4te t + e t 4e t 6, y = 3y + 4(x + z) 4te t 4e t 6, z = 6(x + y) 5z + 6te t + 6e t + 75. x = 2(5x + y + 4z) 8te 2t 2e t + e 2t 28, y = 2(x + z) 2te 2t e t 32, z = 2x 3y z + 2te 2t + 3e t + 6. x = 2(x 2y) + e 2t + 4e t, y = 2(y + 2(x + z)) 4te 2t e t 68, z = 2(z + 3y) + 6e t + 34. 22
8. x = 2(y z) 3x + e t te t 2e t + 36, y = 2(y + x + z) 2te t e t 36, z = 3(x y) + 2z 3te t + 3e t 36. 9. x = 9x + 4y + 8z 8te t + e t 4e t 52, y = y + 4(x + z) 4te t 2e t 76, z = 2x 6y z + 2te t + 6e t + 29. 2. x = 2(4x + y + 3z) 6te 2t + e 2t 2e t 2, y = y + 2(x + z) 2te 2t 4, z = (9x + 3y + 7z) + 9te 2t 3e t + 4. IV. Çíàéòè çàãàëüíèé ðîçâ'ÿçîê ñèñòåìè äèôåðåíöiàëüíèõ ðiâíÿíü, âèêîðèñòîâóþ è ìåòîä âàðiàöi äîâiëüíèõ ñòàëèõ.. x = x + y + z + te t, y = y + z, 2. x = x + y + 2z + te t, y = y + z, 3. x = x + y z + te t, y = y + z, 4. x = x + y 2z + te t, y = y + z, 5. x = x + 2y + z + te t, y = y + 2z, 6. x = x + 2y + 2z + te t, y = y + 2z, 7. x = x + 2y z + te t, y = y + 2z, 8. x = x + 2y 2z + te t, y = y + 2z, 9. x = x + y + z te t, y = y + z,. x = x + y + 2z + te t, y = y + 2z, z = z et 23
. 3. 5. 7. 9. x = x y + z + te t, y = y z, x = x y z + te t, y = y z, x = x 2y + z + te t, y = y 2z, x = x 2y z + te t, y = y 2z, x = x y + z te t, y = y z, Ñïèñîê ëiòåðàòóðè 2. 4. 6. 8. 2. x = x y + 2z + te t, y = y z, x = x y 2z + te t, y = y z, x = x 2y + 2z + te t, y = y 2z, x = x 2y 2z + te t, y = y 2z, x = x y + 2z + te t, y = y z, z = z et. Ñàìîéëåíêî À.Ì., Êðèâîøåÿ Ñ.À., Ïåðåñòþê Í.À. Äèôåðåíöiàëüíi ðiâíÿííÿ ó ïðèêëàäàõ i çàäà àõ. Ê.: Âèùà øê. 994. 455 ñ. 2. Ñáîðíèê çàäà ïî ìàòåìàòèêå äëÿ âòóçîâ..2. Ñïåöèàëüíûå ðàçäåëû ìàòåìàòè åñêîãî àíàëèçà: Ó åá. ïîñîáèå äëÿ âòóçîâ / Ïîä ðåä. À.Â.Åôèìîâà è Á.Ï.Äåìèäîâè à. 2-å èçä. Ì.: Íàóêà, 986. 386 ñ. 3. Ôèëèïïîâ À.Ô. Ñáîðíèê çàäà ïî äèôôåðåíöèàëüíûì óðàâíåíèÿì: Ó åá.ïîñîáèå äëÿ âóçîâ. 7-å èçä., ñòåð. Ì.: Íàóêà, 992. 28 ñ. 4. Ãîëîâà Ã.Ï., Êàëàéäà Î.Ô. Çáiðíèê çàäà ç äèôåðåíöiàëüíèõ òà iíòåãðàëüíèõ ðiâíÿíü. Ê.: Òåõíiêà, 997. 288 ñ. 24