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Maple
. .
, . , 2007 .


1 1.1 . 1.2 . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 Maple . . . . . . . . . . . . . . . . . . . . . 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 2.2.2 2.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 10 12 15 15 16 2 3 6

1


1.
x1 = f (x0 , y0 , ), y1 = g (x0 , y0 , ), (1)

-- , ; , M = (x0 , y) ) M1 = (x1 , y1 ). , . x1 , y1 , , M2 x2 = f (x1 , y1 , ), y2 = g (x1 , y1 , ). (2)

, M = (x0 , y0 ) M2 = (x2 , y2 ). (1) (2). (1) (2) , , x2 = f (x0 , y0 , ), y2 = g (x0 , y0 , ).

, , . (Sophus Lie), 1871.
1

1

Ё Klein F.; Lie S. Uber diejenigen ebenen Curven, welche durch ein geschlossenes System von einfac...//

Math. Ann. 1871. S. 50 - 84

2


1.1. . dy dx = = d. (x, y ) (x, y ) dx dy - = 0, (x, y ) (x, y ) , (3), F (x, y ) = C1 . : = dx = g (x, y , C1 ) + C2 . (x, y (x, C1 )) (3)

C1 = F (x, y ) , , (3) F (x, y ) = C1 , G(x, y ) - = C2 .

, x| F (x, y ) = F (x0 , y0 ), G(x, y ) = G(x0 , y0 ) + . (4)
=0

= x0 ,

y |=0 = y

0

, M0 = (x0 , y0 ) M = (x, y ), . , = 1 M0 = (x0 , y0 ) M1 = (x1 , y1 ), F (x1 , y1 ) = F (x0 , y0 ), G(x1 , y1 ) = G(x0 , y0 ) + 1 ,
3


= 2 M1 = (x1 , y1 ) M2 = (x2 , y2 ), F (x2 , y2 ) = F (x0 , y0 ), (x2 , y2 ), F (x2 , y2 ) = F (x0 , y0 ), G(x2 , y2 ) = G(x0 , y0 ) + 1 + 2 . G(x2 , y2 ) = G(x1 , y1 ) + 2 ,

= 1 + 2 M0 = (x0 , y0 ) M2 =

, , , , = 0 , . 1. , , (4) (3). . , (1) : f (x1 , y1 , ) = f (x0 , y0 , ), g (x1 , y1 , ) = g (x0 , y0 , ),

. = (, ) x0 , y0 -- , -- , -- . , f dy1 f d f dx1 + = , x1 d y1 d d g dx1 g dy1 g d + = . x1 d y1 d d

f g x1 , y1 , x1 y1 dx1 = (, ) (x1 , y1 , ), d
4

dy1 = (, ) (x1 , y1 , ). d


x1 , y1 , , dy1 dx1 = () (x1 , y1 ), = () (x1 , y1 ). d d , (4) dy1 dx1 = = ()d. (x1 , y1 ) (x1 , y1 ) x1 |
=0

(5)

= x0 ,

y1 |

=0

= y0 ,

0 -- , .


=
0

()d.

(4) . , (4), u = F (x, y ), v = G(x, y ),

: u1 = u0 , v1 = v0 + . (6)

, ( , ) A = (x, y ) + (x, y ) . x y

, (x, y ) (x1 , y1 ) = (x0 , y0 ) + A[]|(x + 12 A [] 2! 2 + . . . ,
(x0 ,y0 )

0

,y0 )

(7)

5


A[] = (x, y ) 1.2.

+ (x, y ) . x y

, , . (x, y ), , (x1 , y1 ) = (x0 , y0 ). dx dy = F (x, y ) (4) (x, y ) = (F (x, y )). (7) , , , A[] = 0. H (x, y , ) = 0 (8)

, . , (, ), H (x0 , y0 , ) = 0
6


(4) H (x1 , y1 , ) = 0. , (8) , : 2. , u(x, y ) = A, , A[u] u: A[u] = h(u). . u(x, y ) = , u(x0 , y0 ) = u(x1 , y1 ) = (, ). (7) u(x1 , y1 ) = + A[u]|(x,y u(x, y ) = A[u] = h() (x, y ). , A[u] = h(u).
7
)=(x0 ,y0 )

+ ...,


, u(x, y ) A[u] u h(u), (7) u(x0 , y0 ) = 1 h ()2 + . . . . 2! (, ), , u(x1 , y1 ) = + h() + u(x0 , y0 ) = u(x1 , y1 ) = . , u(x, y ) = -- . (8) M (x, y )dx + N (x, y )dy = 0, H H dx + dy = 0. H = 0, x y . , . 3. , M dx + N dy = 0 A = x + y , 1 µ= M + N .
8




. u(x, y ) = const -- M dx + N dy = 0. 2, , u(x, y ) = , , (x, y ) u u + (x, y ) = h(u). x y

u -- , µ, du = µ(M dx + N dy ), µM + µM = h(u), h(u) M + N , h(u) -- , , (M + N )-1 -- µ= . u = F (x, y ), v = G(x, y )

(6). u0 , v0 du1 = 0, (3) du dv = = d. 0 1 , M (u, v )du + N (u, v )dv , M (u, v ) du + dv N (u, v )
9

dv1 = d,


. , M (u, v ) = 0, v N (u, v ) M (u, v ) = F (u). N (u, v ) , M dx + N dy = 0 , : S (u)du + dv = 0. , . (4), S (F (x, y ))dF (x, y ) + dG(x, y ) = 0.

2.
M (x, y )dx + N (x, y )dy = 0. (9)

, A = x + y , 3 1 ; M + N (9) µ= M dx + N dy = const. M + N
10




(9). 2.1. x1 = x0 , y1 = y
0

, . x0 y0 dx1 = x0 d = x1 d , dy1 = y0 d = y
1

d ,

(5) dx1 dy1 d = = = d. x1 y1 =
1

d = ln ,

= exp . y1 y0 =, x1 x0 ln x1 = ln x0 +

A = xx + y y . , F F F y y d + d ln x = 0, xx y dy - y dx dx + = 0, x x2 x y y dy + (1 - )dx = 0. x x
11


F , dy y =G . dx x ,

, µ = (xG - y )-1 . , x, t = y /x G(t)dx - d(xt) dx dt Gdx - dy = = - xG - y xG(t) - tx x G(t) - t . 2.1. Maple , . , , . 1990- E.S. Cheb-Terrab , DETools Maple.
2

, K . , . M N -- , µ=
2

1 M + N

E.S. Cheb-Terrab, E.S. Cheb-Terrab Computer Algebra Solving of First Order ODEs // Computer

Physics Communications, 101 (1997) 254.

12


µM µN - =0 y x xn y m , . , , , . M N -- , , . Maple way 3 () 4 ( ), K dgun. 236 . , x y : { = 0, = f (x)}, { = 0, = f (y )}, { = f (x), = 0}, { = f (y ), = 0} (10) f -- . . ., = 0, = f (x) y0 y1 = + f (x1 ) f (x0 ) , x1 = x0 , F (x)dx + d y =0 f (x)

dy f (x) = y + G(x), dx f (x)
13


F G -- . y = (x, y ) y , a(x), : = 0, = f (x) = exp a(x).

, y = (x, y ). , , { = 0, = f (x) g (y )}, { = 0, = f (y ) + g (x)}, { = f (x) g (y ), = 0}, { = f (y ) + g (x), = 0} (11)

f g -- . 120 . ( , ) x y Maple, . equinv.mw. DETools . 576 , , , . 47, 48, 50, 55, 56, 74, 79, 82, 202, 205, 206, 219, 234, 235, 237, 265, 250, 253, 269, 331, 370, 461, 503 576, , . 552 , , ґ DETools . , 78% 552 , , . 2000 . E.S. Cheb-Terrab, T. Kolokolnikov3
3

E.S. Cheb-Terrab, T. Kolokolnikov First order ODEs, Symmetries and Linear Transformations //

Submitted to "European Journal of Applied Mathematics" - July 2000.

14


, , , , . , - , , , "" , . 2.2.
2.2.1.

p(x1 )y1 + q (x1 ) = p(x0 )y0 + q (x0 ), f (x1 ) = f (x0 ) + (12)

E.S. Cheb-Terrab . , , , . (x0 , y0 ) , pdy1 + (p y1 + q )dx1 = 0, pf dy1 f dx1 =- = d 1 p y1 + q , = F (x), = P (x)y + Q(x). S (py + q )d(py + q ) + df = 0, S pdy + (f + S (p y + q )dx = 0, dy f (x) p (x)y + q (x) =- G(p(x)y + q (x)) + , dx p(x) p(x) G -- .
15

f dx1 = d,



(13)


2.2.2.

, , dy = (x, y ), dx (14)

(12). , -- , , y
yy

,

. (i) (14) (12), (x, y ) x, u, (14), du = (x, u), dx (x)x + (x)u . , y = f G (py + q ) - yy = f pG (py + q ), y 1 G yy = K (py + q ) K = . yyy p G A(x, y ) := yy . yyy (15)
yy

p , p

= f p2 G (py + q ),

(16)

Ay 0, k , k p(x) = A(x)-1 ,
16


(14) A(x1 )-1 y1 + k q (x1 ) = A(x0 )-1 y0 + k q (x0 ), x, (14), du = (x, u), dx u1 + g (x1 ) = u0 + g (x0 ), f (x1 ) = f (x0 ) + u = A(x)-1 y f (x1 ) = f (x0 ) +

(x)x + (x)u . A
yy

0, Ay 0, A(x, y ) k1 (py + q ) + k0 p (k1 = 0),

(14) k1 [p(x1 )y1 + q (x1 )] + k0 = k1 [p(x0 )y0 + q (x0 )] + k0 , ln A(x1 , y1 ) + ln p(x1 ) = ln A(x1 , y1 ) + ln p(x1 ), x, u = ln A(x, y ) f (x1 ) = f (x0 ) + f (x1 ) = f (x0 ) +

(14) u1 + g (x1 ) = u0 + g (x0 ), f (x1 ) = f (x0 ) +

(x)x + (x)u .
17


, Ayy 0, B (x, y ) := Axy py+q = Ayy p (17)

B y p(x) = k exp By dx,

k -- . x, u = exp By dxy .

: (14) (12), u = a(x)y , u = ln a(x) + b(x)y f (x)x + g (x)u . , u1 + p(x1 ) = u0 + p(x0 ), f (x1 ) = f (x0 ) + .

, a(x1 )y1 + p(x1 ) = a(x0 )y0 + p(x0 ), a(x1 ) + b(x1 )y1 a(x0 ) + b(x0 )y0 = , e-p(x1 ) e-p(x0 ) . : 4 (E.S. Cheb-Terrab, T. Kolokolnikov, 20004 ). (14), . A(x, y ) :=
4

f (x1 ) = f (x0 ) +



f (x1 ) = f (x0 ) + ,

yy yyy

E.S. Cheb-Terrab, T. Kolokolnikov First order ODEs, Symmetries and Linear Transformations //

Submitted to "European Journal of Applied Mathematics" - July 2000.

18


y u : Ay 0 u = A(x)y Ay 0, A
yy

0

u = ln A(x, y )
A

Ay Ayy 0 u = exp By dxy , B (x, y ) := Axy yy (14) f (x)x + g (x)u , (14) (12). (ii) du = (x, u) dx f (x)x + g (x)u , f g , , . (3) du dx = = d, f (x) g (x)
x
1

x

1

x

1

x

0

u1 -

g (x) dx = u1 - f (x)

g (x) dx f (x)

dx = f (x)

dx + , f (x)

S u- g (x) g (x) dx dx (du - + = 0, f (x) f (x) f (x) g (x) dx f (x) , (18)

du 1 = g (x) - G u - dx f (x) G -- . E (x, u) :=
19

u , uu

(19)


E (x, u) = G =E u- G g (x) dx f (x)

Eu 0, (x, u) F (x) := , µ = g (x) - f (x)(x, u) du - dx, 1 + =0 x µ u µ -µ-2 µx - µ-2 µu + µ-1 u = 0 µx + µu - µu = 0 , , g - f - f x - g u = 0. (20), (F + x - F u )f + (F + )f = 0. , k f = k exp - F + x - F u . F + (21) g (x) Ex =- . Eu f (x) (20)

, f = exp - F + x - F u , F +
20

g = -F (x)f ,

(22)


, (x, u). , . Eu 0, Ex 0 (19) k1 , G = k1 , G G(s) = k2 + k3 exp k1 s (18) du = a(x) + b(x) exp k1 u, dx (23)

a b -- , f g . f (x)x + g (x)u , (21) (x, u) = a(x) + b(x) exp k1 u f , g . , , exp k1 u 1: - f b - f b - k1 g b = 0. g - f a - f a = 0. : g = f a + C1 , f b (f b) = -k1 (f a + C1 )b = -k1 a(f b) - k1 C1 b, f b = C2 - k1 C
1

b(x)e

+k1

a(x)dx

dx e-

k

1

a(x)dx

21


, C1 = 0 C2 = 1 f= exp -k1 a(x)dx , b(x) g= a(x) exp -k1 a(x)dx . b(x)

3, µ := f (x) (a(x) + b(x) exp k1 u) - g (x) (23), (23) .
2.2.3.

(14), . . . .

22