例3 设 =e*,求y(n),解 y'=e*,y"=e*, y" =e",..,(e*)(n) =e*例4设 y = In(1 +x)(x >-1),求y(n),11解 131+x(1 + x)2!3!D（4)(1 + x)(1 +x)4J(") =(-1)-i (n -1)!(n ≥1, 0!=1)(1 +x)
例 3 , . x (n) 设 y = e 求y 解 , x y = e , x y = e , , x y = e ( ) . x (n) x e = e 例 4 ln(1 )( 1), . (n) 设 y = + x x − 求y 解 x y + = 1 1 2 (1 ) 1x y + = − 3 (1 ) 2!x y + = 4 (4) (1 ) 3!x y + = − ( 1, 0! 1) (1 ) ( 1)! ( 1) ( ) 1 = +− = − − n x n y n n n
例5 设= sin x,求y(n)T解 J'=cosx = sin(x+π2元元T儿sin(x + 2.y" = cos(x + sin(x +2222元元sin(x + 3.y" = cos(x + 2.22元元y(n) = sin(x+n.即 (sinx)(n) = sin(x + n :22元同理可得(cosx)(")= cos(x+ n.2
例5 sin , . (n) 设 y = x 求y 解 y = cos x ) 2 sin( = x + ) 2 cos( y = x + ) 2 2 sin( + = x + ) 2 sin( 2 = x + ) 2 cos( 2 y = x + ) 2 sin( 3 = x + ) 2 sin( ( ) y = x + n n ) 2 (cos ) cos( ( ) x = x + n 同理可得 n 即 ) 2 (sin ) sin( ( ) x = x + n n