例4. 设y=sin x,求y(n)解： y'=cosx = sin(x+)y"= cos(x +)= sin(x++)= sin(x +2·)y" = cos(x+2·号) = sin(x +3·一般地，(sinx)(n)=sin(x+n·号)类似可证：(cos x)(n) = cos(x + n ·号)oeol00x机动目录上页下页返回结束
例4. 设 求 解: y = cos x sin( ) 2 = x + cos( ) 2 y = x + sin( ) 2 2 = x + + sin( 2 ) 2 = x + cos( 2 ) 2 y = x + sin( 3 ) 2 = x + 一般地 , x = x + n (sin ) sin( ( ) 类似可证: x = x + n (cos ) cos( ( ) ) 2 n ) 2 n 机动 目录 上页 下页 返回 结束
例5.设y=eαxsinbx(a,b为常数),求y(n)解: y'= aeax sin bx + beax cos bx= eax (asin bx + bcos bx)= eax /a? +b? sin(bx+p)(β = arctan =ay"= Va? +b2 [aeax sin(bx +p)+ beax cos(bx +)]= a? +b? eax Va? +b? sin(bx+ 2p)by(n) =(a? +b2)2 eax sin(bx + np) (β = arctan=)aOeo0x机动目录上页下页返回结束
例5 . 设 y e bx ax = sin 解: y = ae bx + ax sin e (asin bx bcos bx) ax = + (a,b为常数), 求 . (n) y be bx ax cos ( sin cos ) 2 2 2 2 2 2 bx a b b bx a b a a b + + + + cos sin ax = e sin( ) 2 2 a + b bx + ( arctan ) a b = 2 2 y = a + b ( ) 2 2 2 ( ) n n y = a + b ax a b e 2 2 = + ( arctan ) a b = sin( 2 ) 2 2 a + b bx + e sin(bx n) ax + 机动 目录 上页 下页 返回 结束