设f(x)满足2f(x)+x^2f(1/x)=x/(x+1),则f(x)=?

Question

设f(x)满足2f(x)+x^2f(1/x)=x/(x+1),则f(x)=?

Answer 1

分式有意义,x≠0,x+1≠0,解得x≠0且x≠-1
2f(x)+x²f(1/x)=x/(x+1) (1)
将x换成1/x
2f(1/x)+(1/x²)f(x)=(1/x)/(1/x +1)
整理,得
f(x)+2x²f(1/x)=x²/(x+1) (2)
(1)×2-(2)
3f(x)=x(2-x)/(x+1)
f(x)=x(2-x)/[3(x+1)]
所求函数表达式为f(x)=x(2-x)/[3(x+1)] (x≠0且x≠-1)