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

Question

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

Answer 1

XY+e^Y=X+1 两边对X求导：
Y+XY'+Y'e^Y=1 解出：Y'= (1-Y)/(X+e^Y) (1)
对（1）两边再对X求导：
Y''= [-Y'(X+e^Y)-(1-Y)(1+Y'e^Y)]/(X+e^Y)^2 (2)
X=0 时,Y(0)=0,Y'(0)=1,
Y''(0)=[-1-(1+1)]/(0+1)^2=-3
即最后得到：d^2Y/dx^2|x=0 ＝ －3.