y=yx由方程e^x+x=e^y+y所确定,求y‘’

Question

y=yx由方程e^x+x=e^y+y所确定,求y‘’

Answer 1

两边对x求导：
e^x+1=y'e^y+y'
得y'=(e^x+1)/(e^y+1)
则y"=[e^x(e^y+1)-y'e^y(e^x+1)]/(e^y+1)²
=[e^x(e^y+1)-(e^x+1)²e^y/(e^y+1)]/(e^y+1)²
=[e^x(e^y+1)²-e^y(e^x+1)²]/(e^y+1)³
=[e^xe^2y+e^x-e^ye^2x-e^y]/(e^y+1)³