ǻСͺԹõ

ǻСͺԹõ ( integrating factor )

Ԩóԧ͹ؾѹѭѹѺ˹ٻ

P(x,y)dx + Q(x,y)dy = 0 (1)

(1) 蹵ç Ҩտѧѹ F(x,y) ͹Ҥٳʹ (1) Ƿŷ蹵ç ¡ѧѹ F(x,y) ǻСͺԹõ ͧ (1) ҧ õ仹

ydx xdy = 0 (2)

2ydy + xdy = 0 (3)

3ydx + 2xdy = 0 (4)

ء蹵ç ͤٳʹô¿ѧѹ 蹵çѧ仹

* ٳʹ (2) y-2

ydx xdy = 0

y2

d ( x ) = 0 蹵ç

( y )

* ٳʹ (3) x

2xy dx + x2 dy = 0

d(x2y) = 0 蹵ç

* ٳʹ (4) (xy)-1

3 dx + 2 dy = 0

x y

d( 3ln |x| + 2ln |y| ) = 0 蹵ç 繵

ѧ鹿ѧѹ y-2 , x (xy)-1 繵ǻСͺԹõҧ

ͤٳ (1) F(x,y)

F(x,y) P(x,y) dx + F(x,y) Q(x,y) dy = 0 (5)

蹵ç Ҥӵͺͧ (5) Ҩդӵͺ͡˹仨ҡӵͺͧ (1) Ǥ Ҩդӵͺ繤ӵͺͧ (5) 繤ӵͺͧ (1) ӵͺѧǹ ͤӵͺͧ F(x,y) = 0 ѧҤӵͺ繤ӵͺͧ ( 1) ҡ (5) Ǥ÷ͺҤӵͺͧ F(x,y) = 0

ҧ 2.1Ҥӵͺͧ y(x3 y) dx x(x3 + y) dy = 0 Ըշ Ѵ

x3(y dx x dy) y(y dx + x dy) = 0 (1)

d ( x ) = y dx x dy ֧ٳʹ (1) y-2

y y2

x3 (y dx x dy) _ (y dx + x dy) = 0

y2 y

x3 d ( x ) _ 1 d (xy) = 0 (2)

y y

(2) 蹵ç Էͧ d ( x / y) 繿ѧѹͧ x / y Էͧ d (xy) 繿ѧѹͧ xy ѧǻСͺԹõٻ xmyn ٳʹ(2)

xm+3 yn d ( x ) xm yn-1 d(xy) = 0 (3)

( y)

͡ m n m + 3 = - n , m = n 1

m = - 2 , n = - 1 ǻСͺԹõ x-2 y-1 (3) ٻ

x d ( x ) _ ( xy )-2 d (xy) = 0

y y

d [ 1 ( x )2 ] + ( xy )-1 = 0

[2 y ]

觨ӵͺ ( x )2 + ( xy )-1 = C ANSWER

2 y

ҵǻСͺԹõ

ҨоԨóҵǻСͺԹõͧԧ͹ؾѹѭѹѺ˹ 蹵çٻ

P (x,y) dx + Q (x,y) dy = 0 (1)

ص F(x,y) 繵ǻСͺԹõͧ (2) ѧ FP dx + FQ dy = 0

F = F(x,y) , P = P(x,y) Q = Q(x,y) 蹵ç

FP = FQ

y x

1 [ QF _ PF] = P _ Q (3)

x y y x

F(x,y) 繵ǻСͺԹõͧ (2)

F(x,y) ʹͧ (2) ҹ




õԵҡ http://158.108.22.9/suchai/417267/variable/work2.htm

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