- ( ) - 3
579 : - - : - / : : 3 4 579-4 5 9 3 9 4 3 5 5 6 3 33 34 3 35 4 36 39 c - ( ) 3 c 3
- - ( ) - ( - ) - - - ( ) - - ( - ) ( t) - dx k = f k (x x n ) k = n () dt x x n f f n - d X = f( X) dt f k (x x n ) k = n D - x x n t X D - X =(t) () (t )= X X = (t) () - (n +)- t x x n t X = (t) x = (t) x = (t) x n = n (t) n- x x n - x = (t) x = (t) ( ato \ -"+ nmoc \ ") - - \ " 4 5
x n = n (t) - R n () - () t D () f( X) D () - f( X) = = ( ) - - D : f() = = X(t) () X(t) d dt = = f() = X(t) f() = f() = d X dt = d dt = X(t) ( ) () () - f( X) = ( ()) X =(t) () c R X =(t + c) X = (t) X = (t) () (t )= (t ) (t) = (t + c) c = t t 3 ( ) X = (t X ) - () X = t= = X (t + t X )= (t X(t X )) = (t X(t X )) - () D - () ( ) () - : ) ; ) () T>; 3) X =(t) (n +)- t x x n \ - x x n ( ) ( - ) = kx k > ( - ) x = x = x - = x () = kx x x ( ) 6 7
x x : x k cos x = C ( u(x x )=x k cos x (3)) 3 x x x x t x x t () kx dx + x dx = kx + +x = C (k >) ( u(x x )=kx + x ()) - : = k sin x k> = x (3) = k sin x =x =ksin x x = n (n = ) x = T = 3 (k ) - ((k +) ) t (k sin x dx + + x dx = ) 3 3 = a x + a x (4) = a x + a x A = a a a a a ij = const (4) P - x x (4) (4) - A - a a a a = ( ) det A - 8 9
: ) A ; ) A -- ( ); 3) A ( = = ) det A ( - ( ) ) I h h t ( +) X = x = C x e t h + C e t h (5) C C R = C e t = C e t (6) h h x = x h + h (7) (7) P h h P - P P X = h + h ( ) (6) (6) - C C = C e t = C e t = C e t = C e t = C e t = C e t (C C ) - - C = C = (6) ( ) C > C = C = C > t (6) = C (8) I < < lim t+ e t = lim t+ e t = X(t) = C e t = C e t ( ) (8) = A > \ " ( 4) - t t + \" h -
p p p (h ) x x (h ) 4 < < 5 > > > > ( 5) 4 5 - P x x I < < e t t+ - - e t t+ + - = A \- " + t + - \" ( 6) 6 < > 7 ( k k>) = x k k > (9) x = x x = - = x - = x kx k 8 k = + k += = k k 4 k> ( ) k = 3 = 3 = 3 - h = 3 h = 3 > > t + \" - h 3
x x h h \ " \ " - ( 7) - ( x = ) 8 ( - ) \ ( 9) p p p II - A : = + i = i - h = + i V h = i V h e t h e t - (4) X = x (t) = C x (t) Re(h e t )+C Im(h e t ) C C R V X = + V = e t (a cos t + b sin t) = e t (b cos t a sin t) = Ce t cos(t ) = Ce t () sin(t ) II : = = i = C cos(t ) = C sin(t ) C > + = C - P V - 9 = i Re < II : = i Re > < () e t t + - = Ce t = t () = () cos = () sin = Ce + - ( ) > - lim t+ =+ ( ) - - 4 5
- 3 ( ) k< ( ) k = 8 5 = 4 5 + i 3 5 = 4 5 i 3 5 ( = 4 5 < ) - ( ) A(a ) a> x = a> x = = x = x kx = A a x ( ) k A x k - x A (k =) - - (4) IIa IIb III = = III h h X = C e t h + C e t h - h h X = h + h = C e t = C e t e t ( < < ) ( > ) 3 4 3 4 p 3 = < 4 = > III h (4) X = x (t) = C x (t) e t h + C e t (h t + h ) () h (A E)h = h h h h h X(t) = h + h = e t (C + C t) = C e t < > - ( 5 6) p 6 7
p p x x x x 7 < = 8 = = 5 = < 6 = > IV det A = a x + a x = a x + a x = det A = IV = h h - X(t) =C e t h + C h h h X(t) = h + h = C e t = C = - ( 7) IV = = IV (4) - IV () = X(t) =C h + C (h t + h ) X = h + h = C + C t = C : = x x = x x = = = h = h = = = const = ( 8) X = A X n =3 a ij = const det A 3 - - - - 8 9
( t +) - ( 3) - ( - ) t + - (85798) - X = f( X) () f k (x x n ) k = n - D X(t) ( f() =) () X t= = X X(t X ) X(t ) = ) U () - X - X(t X ) t ; ) > >(< ) X < X(t X ) < t X X ) ) lim X(t X )= X t - t + 3 > > X X < - t = T> X(T X ) > - ( ) ( ) ) ( Re < Re < ) ; ) ( Re =Re =); 3) ( ) n = Re \" n X = A X (3) X(t) = -
X = () - A X = () - A ( 9) Im 9 Re A - (Re <) X = ; X = - X(t X ) C X e t > \ " ( ) ( - ) 4 4 V ( X) =V (x x n ) - U X = U V ( X) > X V () = V ( X) < X V () = V ( X) 5 V ( X) = V (x x n ) () V ( X) = = x V f + + V x f n f f n f( X) X = f( n X) X = (t) () V ( X(t)) = V ((t)) = V ( (t) n (t)) dv ( X(t)) dt = V d V d n + + x dt x n dt X = (t) d k dv ( X(t)) dt = f k dt = x V f + + V x f n n - ( ) 6 - U X = V ( X) () X = f( X) V ( X) X U 3
X = () X = X = - ( V ( X) < X V () = ) X = n = V ( X) - V ( X) =C - ( ) C C V C x x V ( X) V = C X = V ( X) V ( X) < X lim V ( X) = X t + x x 4 - ( ) = y = y x (I) (II) 3 = x = x y 3 - V (x y) =x + y ( ) (I) V = V x y + V y (x) =xy xy = (II) V = V x (y x3 )+ V y (x y3 )=xy x 4 xy y 4 = (x 4 + y 4 ) (I) - ; (II) - - x = y = - - - 5 () X = f( X) n X = ( f() = ) 4 5
- f( X) - - f( X) X = f() = = f k (x x n )= = f k X= x x + + f k X= x x n + r k (x x n ) n f k (x x n ) r k (x x n ) M k X X ( X = = x + + x n ) () X = A X + R( X) (4) f f x A = x n f n f x n x n X= R( X) M X (4) R( X) X = A X (5) (5) () X = ( ) X = f( X) f() = f( X) X = A - X = X = f( X) - A - X = X = f( X) - - - : - 4 (I) (II) (I) = i (II) = x = k sin x k > = x ( ) - = kx k > = i k - ( 3 ) 5 = ln(5 x y) = e xy (6) - 6 7
ln(5 x y) = v v e xy = : A (; ) A (; ) ) A (; ) x = u y = v + A (u v) u = ln( u v) v = e u+uv ( ) - ( ) ( ) - u = u v (7) v =u ( ) = i 3 (7) - Re( )= < - A (; ) (6) (7) (u v) - B(b ) b> (7) u = b v =: uv = b ( ) - ( ) ) A (; ) x = + y = = ln( ) = e + = (8) = 8 ) B(6) u ) ( ) = = (8) A: ( ) = a = h 4 b = ( ) 4 = a = h b = u e t t+ e t t+ + (8) Re => (6) - A (; ) ( ) 6 =3x 4 x (9) - x = x x = x = x =3x 4x x () 9
x = 3x 4x x = A (; ) A (3; ) ) A (; ) (; ) () x = x =3x 4x ( ) A = 3 4 = = 7 < = + 7 > h = = 7 h 3 = e t 7 t+ e t t+ + - 3 () (9) - (; ) > ) A (3; ) x =3+ x = A = = 3 () 4 = = 3 4 ( ) A = 3 4 = = 3 - (; ) - h = h = = \" - 3 h 4 () - (9) (3; ) Re < Re < (h ) x (h ) x 3 4 6 X = f( X) X t= = X X(t X ) X(t X ) ) U X X(t ) t ; ) > >( ) X < t X(t X ) X(t ) < 3 3
X(t X ) : X(t) = (t)+ X(t X ) (t) (t) = f( (t)+ X(t X )) d X(t X ) = dt = f( (t)+ X(t X )) f( X(t X )) = G(t ) (t) =G(t ) () (t) (t) = G(t ) (t) ( (t ) t ) - ( 3) () (t ) = G(t ) t < G(t )=A(t) + R(t ) A(t) R(t ) M = A(t) - () A(t) a ij = const 7 = x y + x +t =4x3y xy +t ( ) ( ) A = 4 3 = i - 8 x = =t : x = t + A x = t + B - t x x x x - t : x = =(x A) + B ( 5) - x x - - (t x x ) - (x x ) A= B= x 5 A= B= A= B= x 3 33
n =3 ( \ ") X = A X A - 3 3 a ij = const det A - - - - 3-3 ) ) 3) 4) 5) ) ) 3 ) 4 ) 5 ) 6 : 3 7 Re < 3 < - 3 - ( ) 3 3 8 Re < < 3-3 ( ) 3 3 9-3 - - ( ) 3 U X = X = f( X) f() = U U X = V ( X) U V ( X) > V ( X) > X U V ( X) = - U U X = = y + x 9 = x + y3 ( ) - = y = x ( ) - V (x y) =xy(x + y + y ) U x + y <( 3) U U U - x> y> U x = y = V (x y) =xy(x + y ) > U 3 x V x=y = V y=x = V = V t =3x y(y +x 3 )+3xy (x+y 3 )=3xy(x 4 +y 4 ) > U ( ) 4II - = y x (II) = x y 3 - - 4II - ( ) 9 34 35
4 - : ) ; ) () - T>; 3) X = f( X) n = f( X) D X = f( X) n = - - - X = (t) - : - - t + t ( ) - t + ( 3 ) ( ) t t + ( 3) ( ) t () t + ( 3 ) 3 3 3 = y + x( x y ) = x + y( x y () ) x x + y =(x + + y )( x y ) y x x y = x + y : x = cos y = sin d dt = ( ) d dt = () =(t) = ( ) x + y = () = + t ( ) - (t) : = C> << = +Ce t = C> > Ce t 36 37
d dt = ( ) d dt << > = y ( ) - - 3 ( ) x t + (t) - 3 x + y = t + (t) x + y = () - : 985 - : 985 3 - : 97 4 : 984 38 39
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