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) s1 < 0. C1es t , t (. 6.4, ).

, s1 < 0 1 = C1es1t 0, t.

, - , , , , .

) , s1 > 0, C1es1t t (. 6.4, ), .. C1es1t t. , t.

) Im ) y Im y y1 = c1 es1tt y1 = c1 es1tt s1 Re s1 Re t t y ) Im Im y ) s1 s y1 = c1es1tt + c2 es2tt y1 = c1 es1tt s2 Re s2 Re t t y ) Im Im y ) sy1 = csin(t + ) y1 = c Re s2 Re t t . 6.4 :

- ; - ; - - ; - - ; - ; - ) (6.5) - . .

, s1,2 = i, < 0, y1 = C1eS1t + C2eS2t = Cet sin(t + ) (. 6.5, ), , , e 0 t , t.

- , .

t ) > 0. (. 6.4, ), , ,.

et t y1 = C1eS1t + C2eS2t = Cet sin(t + ) ) , (6.5) , .. s1,2 = i, : = Csin(t + ), .. (. 6.4, ).

y1 = C1ei + C2e-i = ) s1 = 0, , .. y1 = C .

Y(T) , . , Y(T) 0 T.

, , Y(T) = ONST, , Y(T) , . , , . - .

, - . . 6.5.

: , s. , . , . i . - ) i ) i s1 s s s s2 s. 6.5 :

- ;

- , , . , .

6.3 6.3.1 . , .

, , , .

, .

n- n :

dy1(t) / dt = a11y1(t) + a12 y2(t) +... + a1n yn (t) + x1(t);

dy2(t) / dt = a21y1(t) + a22 y2(t) +... + a2n yn (t) + x2(t);

...

dyn(t) / dt = an1yn(t) + an2 yn(t) +... + ann yn(t) + xn(t), .

x(t) = {x1, x2,..., xn} .

(. 6.6), t, ().

, .

.

. . - (6.7) :

dy1(t) = f1( y1, y2 );

(6.8) dydt(t) = f2 (y1, y2 ).

dt .

1 , .. . : y1 = 0, y2 = 0, . :

dy1(t) = 0;

dydt(t) = 0.

dt , , , .

2 . .

3 y1 = 0, y2 = 0, .. , .

4 , dy y2(t) = 0, =, y1 (t) = y(t) .

dt 5 dy1 (t) , - , y2 (t) = > 0 y1(t) = y(t) , dt dy1 (t) y2 (t) = < 0 y1 (t) = y(t) .

dt 6 , y2(t) f2(y1, y2) , dy , dy , , .

M0 .

, , .

6.3.2 , , (6.8) t, :

dy2 f2 ( y1, y2 ) =.

dy1 f1( y1, y2 ) , .

.

d y(t) dy(t) a2 + a1 + a0 y(t) = 0, (6.9) dt dt y(t) - ; a0, a1, a2 - .

dy1(t) y(t) = y1(t), = y2 (t), dt d y1(t) dy2 (t), = dt dt (6.9) :

dy1(t) = y2;

dt (6.10) dy2 a1 a = - y2(t) - y1(t).

dt a2 a , dy2 a1 a0 y(6.11) = - -, dy1 a2 a2 y y2 = f(y1, 1, 2), (6.12) I - .

A2 S2 + A1 S + A0 = 0.

- a1 = 0, a0 > 0, a2 > 0: s1,2 = +i;

a=. .

a : a2y1"(t) + a0y1(t) = 0, y1(t) = Asin(t + ), (6.13) y2(t) = y1'(t) = A cos(t + ). (6.14) y1(t) . 6.7.

(6.13) (6.14) , :

2 y1 y+ = 1. (6.15) A2 (A) (6.15) A A. , , (. 6.7, ).

M y2. y1 , y2 - , , - y2 = y y1 ) ) ) i t y M0 A . 6.7 :

- ;

- ; - , .

, (.. , ).

2 - a12 < 40a2; a> 0, 2 > 0, a0 > 0:

S1,2 = - I (. 6.8, ), = -A1/22, = (1/22) - .

a1 - 4a0a (6.9) :

A Y1(T) = E-T SIN(T + ). (6.16) y2 (t) = y'(t) = e-t os(t + + ), (6.17) a = arctg ;.

= a (6.16) (6.17) ( t). , , , y1 y2 , ..

.

.

) y2 = y1 ) y i ) s t y2 y s t . 6.8 :

- ;

- ; - ) y1 ) y2 = y1 ) i s t y2 y s t . 6.9 :

- ;

- ; - 3 - a21 < 40a1;

a0 > 0, 1 < 0, a2 > 0: s1,2 = + .

, .. . (6.9):

y1(t) = et sin(t + ). (6.18) y2(t) = y'(t) = et os(t + + ). (6.17) , , .

, (. 6.9).

, , .. .

4 A21 > 40A2, A1 > 0, 2 > 0, A0 > 0:

a1 - 4a0aas1,2 = ;.

= ; = 2a2 2a , .

(6.9) 1(t) = C1e-s t + C2e-s2t. (6.20) y2(t) =. (6.21) - C1s1e-s1t - C2s2e-s2t 1 2 y2 = s2 y y2 = s1 y1, (6.20), (6.21) s1 = 0 s2 = 0 ( ).

- , (. 6.10). .

y2 = y ) ) 1 ) y i 4 3 t s1 s2 y y 3 t . 6.10 :

- ;

- ; - 5 - a12 > 40a2, a1 < 0, 2 > 0, a0 > 0: s1,2 = .

, . (6.9):

1 y1(t) =. (6.22) C1es t + C2es t ) ) i y1 ) y2 = y1 t s1 y s y t . 6.11 :

- ;

- ; - y2(t) = y'(t) =. (6.23) C1s1es1t + C2s2es2t , .. ( ), .

(. 6.11). 4 1 1, y2 = s1y1 y2 = s2y1. 2 2.

6 - a1 > 0, a2 > 0, a0 < 0: s1 = -1, s2 =.

( a0 = 0 - ).

, .

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