. , .
.
: , , , , .
1.

\(, ) - (,) = (,), (1)
= (,), , . - [1] (1), . .
(1) .

X = (,,) + (,,)-+ (, , ),
V V I I /0 I , , , I /0
X = X + ( ------+ ----+ -----+ ------+
<= dut dux dutt dutx duxx

pt = Dtn - utDtT - uxDtC, px = Dxn - utDx - uxDxC,
P ^ DxP uttDxT utxDx, p ^ DxP utxDxT uxxDxC,
^
Dt = : + ut;+ utt.-+ utx;-+ > Dx = ;+ ux.+ utx.-----+ uxx;-+
cm Cux ut Cux
,
-\ + ptx + pxxlM = -^Vt + Aut{Tt - ) + ^-ux^t + AutTu + AutuxC'u + Vtx +
+ut (Vxu - Ttx - A^t) + ux(Vtu - C,tx) - utiTxu + A^u) + utux ( - Ttu - xu)--ux&u - u ux Tuu - utux^uu - uttTx + utx(Vu - Tt - x + t) - ututx(2Tu--u) - uxuttTu - 2uxutxCu + Vxx + ut(-Txx + AVu - 2ACx) + ux(2Vxu - Cxx) +
+utux(-2Txu - 3ACu) - ux(-Vuu + 2Cxu) - utu2xTuu - uxCuu + utx(-2Tx - Vu+
+ 2Cx) + uxutx(-2Tu + 3Cu) - Aut Tu + ututxTu = 0.
M , (1). ut, ux, utt, utx
ntx - AVt + Vxx =
Vxu - Ttx - AC,t + ATt - Txx - 2AC,x = 0, ntu - Ctx + AC,t + 2Vxu - Cxx = 0,
-Txu - AC,u = 0,
Vuu - Ttu - Cxu - 2txu - 2ACu = 0>
-C,tu + Vuu - 2Cxu = 0> -Tuu = 0>
Cuu Tuu 0) Tx
Cx - Tt + Ct - 2tx = 0,
C'u - Tu = 0> Tu = 0> C'u - 2tu = 0> C'uu = 0.
, t, u.
Vtx - AVt + Vxx = 0> (2)
Vxu - AC,t + ATt - 2AC,x = (3)
ntu - Ctx + AC,t + 2Vxu - Cxx = (4)
nuu = , (5)
Cx - Tt + Ct = 0. (6)
(6) , Tt = Ct + Cx , Ctx + Cxx = 0 (3)

Vxu - AC,x = 0. (7)
(5) n(t,x,u) = A(t,x)u + B(t,x) (2), (4), (7)
Atx - AAt + Axx = 0> (8)
Btx - ABt + Bxx = 0,
At + 2Ax + AC,t = (9)
Ax = ACx. (10)
(10) , A(t, ) = AC(t, x) + C(t), Atx + Axx =
0. (9)
2ACt + 2ACx + C(t) = 2At (t) + C(t) = 0.
= 0, = + , = () (8). -
, = 0, = + . (,) = -1( + ()). (6) 2 = '(), () = 2 + ,
, 2 + 2 +
() =---^ + , ,) =---------------------------
= + + (, ),
.
, , : = 1, = 2, = , , = . (11).
1. (1) X = I 2 = . = . X = I + (2 ) I. = (.) &. (11)
.
, , 4.
1. (2)-(6) , = 0 (1)
( \
= ()( - + , 2 = = , 4 = ( ) ,
\ )
5 = ( '^ 5 = (
, , , .
2.
(1)

1 = 7 > 2 = 7ғ; 3 = ~ ^ 4 = ^ + (2 )~ ~. (12)

,
14 = 1^ 14 = 2 24 = 1, 24 =
1 __ _ 1 2 _ _ 2 2 _ 1 3 _
41 1 41 2 42 1 42 -
= ^1 -
:
1 = 4 - + 24 -~2,
1 2
(12)
1 2 3 4
1 0 0 0 1 + 22
2 0 0 0 2 3
3 0 0 0 0
4 1 22 2 + 3 0 0

4 \ 4
2 -7,
2 3
- - (21 - 2) + 2
1 ( ) 2 + 3'
, :
-1 1 I 4
1 = 1 + 41,
2 2 + 2 41,

3
24
= 2, 3 42,
1
1 -,
2 1 + (2 1) , 3 1 - + ( 2 1) + 3 ( 2 21).
(13)
(14)
(15)
, -
4
X ^ X^.
=1
(1 ,2 ,3). (15). 4 2 1,
1 1 -,
4 4 , 3 1 (1 -) + 4( 1) + 3'
1 0. 2 0, (3). 2 0, (2 + 3).
1 0. 1 2 (1 + 2 + 3). 1 2 1 2 1 (1 + 3) (1 + 22 + 3).
11 + 22 + 33 + 4, 2 0, 3 0 (13), (14) 2 3 0. 1 0 (4), 1 0 1 (15) , , 1 ^ 1, 2 ^ 2, 3 ^ 3, (1 + 4).
2 3 0, (13), (14), 1 2 3 0 (4).
, .
2. (12) (3), (4), (1 + 4), (1 + 3), (2 + 3), (1 + 2 + 3), (1 + 22 + 3).
. 3 , 4 (,) (,) ( 2)(2). (1)
''() + ' () + 2() 0,
5 2. ( 2) (), 2\/ 2,
2 " + V + 2 0,
0 [2]: () ^() + 20 (),
2
=0
(1)
4(!)2 ,
= ) (1 |+)- ! (|) (1+2+ +

(,) = (-24) (/^/ - 2) + 2(1\/ - 2)) .
1 + 4
\
(1 + () + (| - ) - ^)/ =
,
/ = (1 + ) - 2, </2 = (1 + )-2(2*--1)

= (1 + )2-(2*--1)^((1 + ) - 2) = 2 1(1+*)-(2*--1)^((1 + ) - 2).
(1),
( + \) () + (2 + 1)^; () + 2^() = 0,
= (1 + ) - 2. () = 2<(), = 2/ + 1, 2"() + '() + (2 - 42)() = 0. 2 [2] () = 1/2() + 2^2().
(,) ^(+)-)2 + 1 ^ (2* 1) (1^2(2\/(1 + ) 2 + 1) +
+22(2\/(1 + ) 2 + 1 )^ ,
1 + 3
+ I /2+4 +\/2+4 \
(, ) = ѳ ѳ 2--------- + 2 2-------- , 2 + 4 > 0,
(, ) = ( 2) (1 + 2), 2 + 4 = 0,
/, -) I ^ . /2 + 4 ^ \/2 + 4 \ 2 ,,
(, ) = ( 2 ) ( 1 -----------------------^- + 2 ----------------------^- ) , + 4 < 0.
2
2 + 3 = (,) =
˗+. 2 + 3 .
1 + 2 + 3
2 ( ) |
(, ) = + , = , = .
, 1 + 22 + 3
, ѳ ( +2^/( + 2)2+8 (2 ) -/ +2 + \^ ( + 2)2 +8 (2
(,) = ѳ 4 (2-) + 2 4 (2-) \
( + 2)2 + 8 > 0,
(,) = -+ (2-)+ѳ (ѳ + 2(2 )), ( + 2)2 + 8 = 0,
(,) = -^(2-')+' (ѳ 8(2 + 2)2 + 8+
^ (2 )/( + 2)2 + 8 2
+2 ----------------------------------- , ( + 2)2 + 8 < 0.

1. , . . / . . . . : , 1978. 400 .
2. , . . / . . . . : , 1971. 288 .