Question

设y=y(x)是由方程xy+e^y=y+1所确定的隐函数,求d^2y/dx^2 x=0?

Answer 1

问：设y=y(x)是由方程xy+e^y=y+1所确定的隐函数,求d^2y/dx^2 x=0?

答：xy+e^y=y+1 （1）求 d^2y/dx^2 在x=0处的值：（1）两边分别对x求导：y+xy' + e^y y' = y' y/y'+x+e^y = 1 (2)(2)两边对x再求导一次：(y'y'-yy'')/y'^2+1+e^y y'=0y'^2-yy''+y'^2+y'^3e^y=0-yy''+y'^3e^y=0y''=y'^3e^y / y (3)x=0 时：e^y0=y0+1 //: 由(1)由（2）的前一式y0+e^y0 y'0=y'0 y0+(y0+1)y'0=y'0 y0+y0y'0=0 y'0=-1y''(0)=-e^y0/y0 //:由（3）x+e^y/y=1+1/y 由(1)得来e^y0/y0=1+1/y0y''(0)=-(1+1/y0)