若f(x)满足2f(x)+x²f(1/x)=(3x³+x²+4x+3)/(x+1)，求f(x)．

若f(x)满足2f(x)+x²f(1/x)=(3x³+x²+4x+3)/(x+1)，求f(x)．

令u=1/x，则x=1/u． 2f(x)+x²f(1/x)=(3x³+x²+4x+3)/(x+1) (1) 变为 2f(1/u)+1/u²f(u)=(3/u³-1/u²+4/u+3)/(1/u+1)=(3/u³-1/u+4+3u)/(u+1) 即 2f(1/x)+1/x²f(x)=(3/x²-1/x+4+3x)/(x+1) (2) (1)×2-(2)×x² 解得 f(x)=(x³-2x³+3x+1)/(x+1)