九 - A 1 1 dy 2 xy 3 y 5 dx 2 y 2 5 y 4 x 7 0 3 y'2 y 2 2...

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习 题 九 A 1 .确定下列微分方程的阶数; 1 5 3 2 y xy dx dy 2 0 ) ' ( 5 ) ' ' ( 7 4 2 x y y 3 x e x y y y x sin 2 ) ' ( 2 ' ' ' 5 2 2 . 解:( 1 )一阶 2 )二阶 3 )三阶 2 .验证下列函数是相应微分方程的解,并指出是特解还是通解 . 其中 2 1 , , C C C 是任意常 2 1 , , 是常数; 1 0 4 ' ' , 2 sin y y x y 2 0 ' ' , sin cos 2 2 1 y y x C x C y 3 2 1 2 1 2 1 2 1 0 ' ) ( ' ' , 2 1 ,( y y y e C e C y x x ); 4 0 ' 2 ' ' , 2 2 1 y y y xe C e C y x x 5 0 9 ' ' , 3 y y Ce y x 6 x x x e y y y e x e y 2 ' 3 ' ' , ) 2 ( 3 2 . 解:( 1 )特解 2 )通解 3 )通解 4 )通解 5 )通解 6 )特解 3 .求下列微分方程的解: 3.(1) ) 1 ( 1 d d 2 2 x y y x y 2 y xy y y x 3 2 3 0 d 1 d 1 2 2 x y y x 4 0 d ) 3 2 ( d ) 2 ( x y x x y x 5 0 d ) 6 4 ( d ) 5 3 ( y y x x y x 6 ). 0 ( d 4 d 2 d 2 2 2 x x y x x y y x 解: 1. ) 1 ( 1 2 2 x xy y dx dy 变形得: 2 2 1 1 x x dx dy y y 两端分别积分
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dx x dy y y 2 2 x 1 x - 1 1 c x x y ln 1 ln 2 1 - ln 1 ln 2 1 2 2 2 2 2 . 1 1 - x c x y )( 2. c x y y x cx y x dx y ydy x xy y dx dy ln ln 2 1 2 3 1 1 ) 1 ( 1 ) 1 ( 1 2 2 2 2 2 2 2 3 dx x y dy dx y dy x 2 2 2 2 1 1 1 0 2 1 1 c x x y ) 1 ln( arcsin 2 4 0 ) 3 2 ( ) 2 ( dy y x dx y x 0 , 0 x 2 0 0 y x Q y p dy y x dx y x y x y x u ) 3 2 ( ) 2 ( ) 0 , 0 ( ) , ( ) , ( c y x x 2 2 2 3 4 2 1 c y xy x 2 2 2 3 4 2 1 : 通解为 5 x y x y dx dy y x y x 6 4 5 3 6 4 5 3 dx du x u dx dy ux x y . du u u u c x dx du x u u u u 1 3 2 3 2 ) (ln 2 3 6 4 6 4 5 3 2 2 u u dx du x u 6 4 5 3 则方程可化为 du u u u c x dx du x u u u u 1 3 2 3 2 ) (ln 2 3 6 4 6 4 5 3 2 2 6. ) 0 ( 4 2 2 2 2 x dx y x ydx xdy
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2 ) ( 4 1 x y x y dx dy x u u y ux y u x y '. ' . . 2 4 1 ' u u x u u 经计算可得 0 4 1 2 2 x c cy 4 .求列下微分方程初值问题的特解: 1 ; 2 ) 4 ( , 0 4 y x dy y dx 2 1 ) 0 ( , 0 e -x y dy y xdx 3 0 y(1) 0), (x 0 d d ) ( 2 2 y x x y x y 解:( 1 ydy xdx 4 c x y 2 2 2 1 2 2 ) 4 ( y 特解 2 1 ) 1 ( 2 1 2 x e x y 16 c
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