Let f(x) be differentiable on the interval (0, ) such that f(1)=1 , and _ t x t^2 f(x)-x^2 f(t) t-x =1 for each x0 . Then, f(x) is

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Let f(x) be differentiable on the interval (0, ) such that f(1)=1 , and _ t x t^2 f(x)-x^2 f(t) t-x =1 for each x>0 . Then, f(x) is

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Correct answer: A. 1 3 x + 2 x^2 3

_ t x t^2 f(x)-x^2 f(t) t-x =1 aligned & & x^2 f^ (x)-2 x f(x)+1 & =0 & & x^2 f^ (x)-2 x f(x) (x^2 )^2 + 1 x^4 & =0 & & d d x ( f(x) x^2 ) & =- 1 x^4 & & f(x) & =c x^2+ 1 3 x also f(1)=1 c= 2 3 . & Hence, & f(x) & = 2 3 x^2+ 1 3 x aligned

Let f(x) be differentiable on the interval (0, ) such that f(1)=1 , and _ t x t^2 f(x)-x^2 f(t) t-x =1 for each x>0 . Then, f(x) is

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