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. 5
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?=
? ,
?2 + 1

m(?2 + 1)? ? ?2 ?2
?? + C = ? ln ? + 2? arctg ?
??
m(?2 + 1)? ? ?2 ?2
?+?
? ln (? = ±1).
?2
2ut?mx2
? ln x2 + x2 ,
x2
? x2 +x2 3 , ? — 2? arctg 1 2
x1
1 2
(1.1).
4) P1 , P2 , P4 :
v
u = ?(x3 ) ? mt, ?2 = 2m2 ; u = (?1 ± 2)mt + C.
?
5) J12 , P3 , P4 :
1/2
u = ?(?) ? mt, ? = x2 + x2 ?2 = 2m2 ;
, ?
1 2
v 1/2
u = ± 2m x2 + x2 ? mt + C.
1 2

6) G3 , J04 , P1 :
1 m m
?(?) + x2 , ?2 = 2m?; (x2 + C)2 + x2 .
u= ? = x2 , ? u=
2t 3 3
t 2t
– 15

7) G3 , J04 , J12 :

1 m 1/2
?(?) + x2 , ? = x2 + x2 ?2 = 2m?;
u= , ?
2t 3 1 2
t
2
m
+ x2 .
2 + x2 + C
u= x1 3
2
2t

8) G1 , G2 , J04 :

1 m2
x1 + x2 , ? = x3 , ?2 = 2m?;
u= ?(?) + ?
2
t 2t
m2
x + x2 + (x3 + C)2 .
u=
2t 1 2


9) J03 , J04 , J34 , J12 :

1 m 1/2
?(?) + x2 , ? = x2 + x2 ?2 = 2m?;
u= , ?
2t 3 1 2
t
2
m2 2 + x2 + C
u= x+ x1 .
2t 3 2



10) J12 + ?D, P3 , P4 (? > 0):

x2
x2
u = e? arctg x1 ?(?) ? mt, ? = ln x2 + x2 ? 2? arctg ,
1 2
x1
4(1 + ?2 )?2 ? 4?2 ?? + ?2 ?2 ? 2m2 e? = 0;
? ?
v
1
1/2 1/2
? mt, m = ± , u = 2m x2 + x2 ? mt,
u = x2 + x2
1 2 1 2
2

11) J03 , J04 , J34 , D :

x2 mx2 x1
1 3
u= ?(?) + , ?= ,
2t 2t x2
m2
? 2 (? + 1)?2 + 4??? + 4?(? ? m) = 0; x1 + x2 .
? ? u= 3
2t

12) J04 ? D + 2T, P1 , P2 :

v 21
t1/2
m m
v ln t, ? =
u = ?(?) ? , ? + 3 ? ? 2m 4 = 0;
2
? ?
x3 ? ?
2
m2 m
u = x3 + v (?2 ± 2) ln t±
4t 22
? ?
v 4
2 + 4 2t ? x ? 32t2 v
? ?
x3 v
3
? ?
± m? x2 + 4 2t ? x3 ? + C.
+ 2 ln
? ?
v 3
2
2 + 4 2t ? x
16t x3 3
16 .. , .. , ..

13) J12 , J13 , J23 , J04 ? D + 2T :

v
x2 + x2 + x2
m
u = ?(?) ? v ln t, ? = 1 , 4? ?2 ? 2m? ? ? 2m2 = 0;
2 3
? ?
t
2
v 3
m m1
u = ? v ln t + 2 + 2 2? + ?
? +
43
2
v
v v
2
2
?2
2 + 2 2? + ? ? 2 + 2 2? + ? +
+ ?
2
v v v
? 2 + 2 2? + ? + 2 + C,
+ 2 2 ln
v
v v
m m2
u = ? ln t ? ? 2 + 2 2? + ? + 2 ?
ln
2 4
v
2
? v v + C.
? 2 + 2 2? + ? + 2

14) AO(3) ? S1 + T1 + ?M1 (? < 0):

v v
mtx 2 x2
u = ?(?) + + 2?m arctg ( 2t), ? = 2 ,
2t2 + 1 2t + 1
m2 (? + 2?) + 2? ?2 = 0;
?
v v
mtx 2
+ 2?m arctg ( 2t) ±
u= 2
2t + 1
v ?x 2 (x 2 + 2? (2t2 + 1))
x 2 + 2? (2t2 + 1)
v
± m? 2 arctg ? + + C.
x2 2? (2t2 + 1)

15) S1 + T1 , J12 , Z1 :

mt x2 + x2 + x2
x2 + x2 + x2 x2 + x2
1 2 3
1 2 3
? = 1 2 2,
u= ?(?) + ,
2+1 2+1
2t 2t x3
2?(? + 1)2 ?2 + 2?2 + m2 = 0.
?

16) AO(3) ? S1 + T1 + ?Z1 :

v
x2 mtx 2 2t2 + 1
? 2? arctg ( 2t),
u= 2 ?(?) + 2 , ? = ln
x2
2t + 1 v 2t + 1
2?2 ? (4? + 2 2m?)? + 2?2 + m2 = 0.
? ?

17) S1 + T1 + J12 , Z1 , H1 + P2 :
v
v x2 ? x2 2t + x1 x2 2t2 ? 1
x2 tx 2 1 2
u= v 3
?(?) + m +2 ,
2
2t2 + 1
2(2t2 + 1) (2t2 + 1)
v 2
x1 + 2tx2
, (? + ? 2 )?2 ? 2??? + ?2 + m2 (4? + 1) = 0.
?= ? ?
2 + 1) x
(2t 3
– 17
v v
2P3 , H2 ? P1 ? 2H3 (? < 0):
18) S1 + T1 + 2J12 + ?M1 , H1 + P2 +
v
v t 2t2 ? 3 2 1 ? 6t2
x1 ? x2 + 2 x1 x2 ?
2 2 2
u = 2?(?) + 2mt? + m
(2t2 + 1) (2t2 + 1)
v
v v
2 2 2t
?2 x1 x3 ? 2 x2 x3 + 2m? arctg ( 2t),
2t + 1 2t + 1
v
2 2tx1 + 2t2 ? 1 x2 + v2 2t2 + 1 x3
1
4m2 ? + 12m2 ? 2
, ? =? 2
?= ? ;
3/2 3
(2t2 + 1)
v
t 2t2 ? 3 2 1 ? 6t2 2
x1 ? x2 + x1 x2 ? 2 x1 x3 ?
2 2 2
u = 2mt? + m 2 2 2t + 1
(2t2 + 1) (2t2 + 1)
v
v v v
2 2t 3
?2 x2 x3 + m 2? arctg ( 2t) ± 2 2m arcsin ? + C.
|?|
2t + 1

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