1、null兰州教育学院学报null 2006年第 4期恰当方程通积分公式的应用*时娟( S , 730020)K 1 本文给出了恰当方程的另一个通积分公式,并通过证明和举例比较说明了两个公式的不同应用b1 o M 恰当方程;通积分;原函数;通积分公式 m s | O175 D S M A1null Z Y s T Z M (x, y )dx +N (x, y)dy = 0 ( 1) nullMnully ! nullNnullx ( 2)H ,Z (1) Z ,N Z Y s :xx0M (x, y)dx + yy0N (x0, y)dy = c ( 3)C N Z 6 B Y s T : xx
2、0M (x, y0 )dx + yy0N (x, y)dy = c ( 4) / :! Z ( 1) Z ,5 i f U(x, y), P dU(x, y) = M (x, y )dx + N (x, y )dy = nullUnullxdx + nullUnullydy nullUnullx =M (x, y ), nullUnully =N (x, y) = T :U(x, y) = yy 0N (x, y)dy + null(x) ( null(x) x i V f )1 P U(x, y ) nullUnullx =M (x, y ), | a null(x), P nullUnul
3、lx =nullnullx yy0M (x, y )dy + null#(x ) = M (x, y) M s ( 2) T , T : yy0nullM (x, y)nully dy + null#(x) = M (x, y ) M (x, y ) -M (x, y0 ) + null#(x ) =M (x, y)56* l 2006- 05- 10T e 时娟( 1966- ),女,回族,河北秦皇岛人,兰州教育学院理科部副教授b | null#(x ) =M (x, y0 ) s null (x) = xx0M (x, y0 )dx + c | c = 0y N f U(x, y ) =
4、xx0M (x, y0 )dx + yy 0N (x, y)dy p f by N Z ( 1) Y s : xx 0M (x, y0 )dx + yy0N (x, y)dy = c2null T 1 p Z : yx dx+ (y3 + lnx)dy = 0 : nullMnully = 1x , nullNnullx = 1x Z Z null | x0 = 1, y0 = 0, Y s T ( 4)Z Y s :x1 0dx + y0 (y3 + lnx)dx = c 14y4 + y lnx = c Y s T ( 4) p 1 Y s T ( 3)9 e null 2 p Z : 3
5、x2 + y2y2 dx-2x3 + 5yy3 dy= 0null :y nullMnully = - 6x2y3 , nullNnullx = -6x2y3 Z Z , | x0 = 0, y0 = 1, Y s T ( 3)5 Z Y s :x03x2 + y2y2 dx + y1 (-5y2 )dy = c x3y2 + x+5y = c Y s T ( 3) p 1 Y s T ( 4)9 e null 3 p Z : 2x (1+ x2 - y)dx- x2 - ydy= 0 : nullMnully = - xx2 - y, nullNnullx = - xx2 - y Z Z null | x0 = 0, y0 = 0, Y s T ( 4)5 Z Y s :x0 2xdx - y0 x2 - ydy = c x2 + 23 (x2 - y ) 32 = c E Y s T ( 3) p Z null3null Y s T ( 3) ( 4) V Z p b H T ( 4)1 T ( 3) p 9 1 e ; H T ( 3) E p null H V I n P Y s T ( 4), V | l r T b 3 I : y Z 57
