<<

. 6
( 135 .)



>>

v v
2P3 , H2 ? P1 ? 2H3 (? ? R):
19) ?Z1 + S1 + T1 + 2J12 , H1 + P2 +
v
v2 v v 2t 2t2 ? 3
x2 ? x2 +
u = 2? ?(ln ? + ? arctg ( 2t)) + 2m 1 2
2
2 (2t2 + 1)
v
1 ? 6t2 2 2t
2 x1 x2 ? 2t2 + 1 x1 x3 ? 2t2 + 1 x2 x3 + 2mt? ,
2
+
2 + 1)
(2t
v
2 2tx1 + 2t2 ? 1 x2 + v2 2t2 + 1 x3
1
?= ,
3/2
(2t2 + 1)
3?2 + 4(m? + 3?)? + 12 ?2 + m2 = 0.
? ?
20) J04 + D, H1 + P3 , H2 + ?P2 + ?P3 (? > 0, ? ? 0)
v
2
?x2 x1 m2 m2
u= v + v + x3 ?(?) + x1 + v x2 ,
2
2t ? ? 2( 2t ? ?)
2t
2t
?2
1
+v + 1 ?2 = 0;
m? + 2
?
( 2t ? ?)
2
2t 2
2
?x x
v 2 + v 1 + x3 ?
u=
2t ? ? 2t
v
mt(2t ? 2?)
v v v+
?
4t3 + (2mC ? 2 2?)t2 ? (2 + 2? 2 + 2m?C)t + 2?
v
m2 m2
+ x1 + v x2 .
2
2( 2t ? ?)
2t

§ 6. p(u) = f (t, x)?(?) + g(t, x), ? = ?(t, x, u)
1) G3 , P1 , P2 :
mCx2
m2
? = ut ? ? ? ? ? = 0; 3
u = ?(?), x, ? u= .
2(Ct ? 1)
23
18 .. , .. , ..

2) G1 , G2 , G3 ? J12 :
m2
? = ut ? ? ? ? ? = 0;
x1 + x2 + x2 ,
u = ?(?), ?
2 3
2
mC x2 + x2 + x2
1 2 3
u= .
2(Ct ? 1)

3) J01 , J02 , J03 , J12 , J13 , J23 :

u = ??(?) + mt, ? = u2 + m2 t2 ? x2 ? x2 ? x2 ,
1 2 3
(2?2 ? 2?)?2 ? 2?? + 1 = 0;
? ?
v v
± 2 ± 10
1/2
u = mt ? C u2 + m2 t2 ? x2 ? x2 ? x2 , C= .
1 2 3
2
4) G3 + 2T, P1 , P2 :

4
? = u3 ? 3m2 x3 u + 3m3 t,
u2 = ?(?) + 2m2 x3 , ??2 =
? ;
9
2/3
u2 ? 2m2 x3 ? C + u3 ? 3m2 x3 u + 3m3 t = 0.

5) G1 , G2 ? P2 , P3 :
v v v
2 22 u 2
u+1 t? x2 ? u + 1 x2 ,
u = ?(?), ?= 1
m 2 2m m
m
? ? ? ? ? v = 0;
?
2
v v v
2 22 m 2 m
+ v = 0.
u+1 t? x? u + 1 x2
u+C
2 2 2u 1
m m 2

6) G3 , J04 , P1 :

u = e?x2 ?(?), ? = 2ut ? mx2 , 4m???2 ? 4m? ? ?2 = 0;
? ?
3
u 1 ?
?2
x2 + ln 1+ + C = 0,
1+ m ?1 m
? m
1+ m ?1
?
u 1 ?
?2
x2 + ln 1+ + C = 0.
? m m

7) G1 , G2 , J04 + P3 :

u = e?x3 ?(?), ? = 2ut ? m x2 + x2 , 4m? ?2 ? 4m?? ? ?2 = 0.
? ?
1 2

8) J04 + ?D, J12 + ?D, P3 (?2 + ? 2 = 0):

x2 + x2 u x2
? ln x2 + x2 ,
u= 1 2
?(?), ? = ? ln + 2? arctg 1 2
t t x1
?m(?2 ? ?2 ?2 ) + 2?[?2 + (? 2 + 1)?2 ? 2??] = 0.
? ? ?
– 19

9) G3 , J04 + ?D, P1 :
2ut ? mx2
?+1
3
u = x2 ? ?(?), ?= ,
2
x2
2
?+1 ?+1
4?(? ? m)? + 4m ? 4
2
?2 = 0;
? ? ?? +
?
? ?
2
mx2 + 2Ct ± (mx2 + 2Ct) ? 8mCt (x2 + x2 )
3 3 2 3
(? = ?1).
u=
4t
10) J12 , J34 , D :
x2 + x2 mx2 mt + u
1 2 3
u= ?(?) + , ?= ,
2t 2t x2 + x2
1 2
(?2m2 + ? 2 )?2 + 4?(m ? ?)? + 4?(? ? m) = 0.
? ?
11) J04 ? 2D, G3 + 2T, P1 :
2
u3 ? 3m2 x3 u + 3m3 t
2 2
u = x2 ?(?) + 2m x3 , ? = ,
x3
2
9?(? ? 4m ?)? ? 6??? + ? + 4m = 0.
4 2 2 4
? ?
12) J04 + D + M, G3 , P1 :
v
v
x2 2 u 2
ln 2 ? (t ? 2 ln x3 ),
3
u = x2 ?(?), ?= +
2
u m x3 m
v2
2? ? m?? + m?3 = 0;
? ?
v v v
m2 ? 2 2m? ? m
2 22
? ? ? + C = 0.
v v
ln
m 2 ? 2 2m? + m 2 ? 2 2m? ? m
m m
13) J12 , J04 ? 2D, G3 + 2T :
u3 ? 3m2 x3 u + 3m3 t
1/2
2
x2 x2 2
u= + ?(?) + 2m x3 , ?= ,
1 2 3/4
(x2 + x2 )
1 2
(36m ? ? 9? )? + 12??? ? 4(? + 4m ) = 0.
4 2 2 2 4
? ?
14) J12 + J34 , J13 ? J24 , J23 + J14 :
2
u
u = ?(?) ? mt, ? = t? 8? ?2 ? 2m2 = 0.
+ 2 x2 + x2 + x2 , ?
1 2 3
m

§ 7. p(t, x, u) = ?(?), ? = ?(t, x, u)
1) P1 + K1 + 2J03 , P2 + K2 + 2J04 , J12 + J34 :
2
2ut ? m x2 + x2 + x2 + 1 ? 4 x2 + x2
1 2 3 1 2
= ?(?),
2
[2ut ? m (x2 + x2 + x2 ? 1)]
v
1 2 3
(mt + u) tu ? m x1 + x2 + x2 + 1 + 2 2x1 x3 m2 + 2mx2 (mt ? u)
2
2 3
v
?= ,
2
2 [2ut ? m (x2 + x2 + x2 ? 1)]
1 2 3
(16? 2 ? 1)?2 + 2(5? ? 1)? ? + 16?(? ? 1) = 0.
? ?
20 .. , .. , ..

2) P0 ? K0 , J12 , J34 :
2
2ut ? m x2 + x2 + x2
2m2 x2 + (mt ? u)2 1 2 3
3
= ?(?), ? = ,
2 (x2 + x2 ) 2 (x2 + x2 )
2m m
1 2 1 2
(? + 4)? ? 4??? ? 4?(1 + ?) = 0.
2 2
? ?

3) AO(3) ? P0 ? K0 :
v

<<

. 6
( 135 .)



>>