1. () , , , .
,
x / (t) =f (t, x(t),x(t-t)),t>0 (1)
18 ( 1771), 20 .
(1) x(t) (1) t>t0 , x(t)=j(t) t-t£ t £ t0, j(t)- , .
t0-t£ t £ t0, , ; t0 . ,
X (t0+0)=j(t0). (. .1)
.1
n- L ,
,
tI(t)³0. x(k)(t-tI(t)) k- x(z), z = t-tI(t). , tI(t) .
(2) (t); l=m0-m, m=max .
.
. , l>0, ( , , ). ()
, l=0, ( , , ). ()
, l<0, ( , , , , ). ()
, ...
:
1) - .
2) - .
3) .
x / (t)=f (t, x (t), x (t-t)) (1)
x (t)= j(t), t j(t)- , , t- , t>0( t : t=t(t)).
( ) (1) x(t), (1) x (t)= j(t), t .
(1) ( , . .), , x(t)
x /(t)=f (t,x (t),j0(t-t)), t0£ t £ t0+t, x(t0)=j0(t0),
t0£t£t0+t t-t
[t0-t;t0] , , x(t-t) f j0(t-t).
|
|
x =j1(t) [t0;t0+t], :
x/(t)= f(t,x(t),j1(t-t)), t0+t £ t £ t0+2t, x0(t0+t)=j1(t0+t)
..
x/(t)= f(t, x(t),jn(t-t)), t0+nt £ t £ t0+(n+1)t, x0(t0+nt)=jn(t0+nt)
2. x(t)= 6x(t-1), x=t 0 £ t £ 1 x(t) 1<t£3.
, : x/(t)=6(t-1), 1 £ t £ 2, x(1)=1, x(1)=1. , :
x(t)= ( 1=3(1-1)2+ =1). 2 £ t £3: x/(t)=6 [3(t-2)2+1], x(2)=4, 3(2-1)2+1=4, :
x(t)=6(t-2)2+6t-8.
, j f , , k t0+(k-1)t, .
, t0 x/(t) , , ,
x/(t)=f(t, x(t),j0(t-t)), t0 £ t £t0+t,
x(t0)=j0(t0),, , , ,
x/(t0+0)=j/0(t0-0).
, , t , .