2f(x)+x^2*f(1/x)=(3x^3-x^2+4x+3)/x+1注意有括号哦

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2f(x)+x^2*f(1/x)=(3x^3-x^2+4x+3)/x+1注意有括号哦
2f(x)+x^2*f(1/x)=(3x^3-x^2+4x+3)/x+1
注意有括号哦

2f(x)+x^2*f(1/x)=(3x^3-x^2+4x+3)/x+1注意有括号哦
2f(x)+x^2*f(1/x)=(3x^3-x^2+4x+3)/(x+1)
用1/x替换上式子中的x
2f(1/x) + f(x) /x^2 = (3/x^3 - 1/x^2 + 4/x + 3)*x/(x+1)
= (3 - x + 4x^2 + 3x^3)/[(x+1)x^2]
整理
f(x) + 2x^2 f(1/x) = (3 - x + 4x^2 + 3x^3)/(x+1)
与原方程联立,消 f(1/x)
f(x) + 2x^2 f(1/x) = (3 - x + 4x^2 + 3x^3)/(x+1)
4f(x) + 2x^2*f(1/x) = 2(3x^3-x^2+4x+3)/(x +1)
两式子相减
3f(x) = (6x^3 - 2x^2 + 8x + 6 - 3 + x - 4x^2 -3x^3)/(x+1)
= (3x^3 - 6x^2 + 9x +3)/(x+1)
所以
f(x) = (x^3 - 2x^2 + 3x +1)/(x+1)

将1/x当成x代入：
2f(1/x)+f(x)/x^2=(3/x^3-1/x^2+4/x+3)*x+1
2f(1/x)+f(x)/x^2=3/x^2-1/x+5+3x
等式两边同乘以x^2
2x^2*f(1/x)+f(x)=3-x+5x^2+3x^3
原式×2-上式：
3f(x)=(6x^2-2x+10+6/x)-(3-x+5x^2+3x^3)
f(x)=(-3x^4+x^3-x^2+7x+6)/(3x)

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