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 equal to

Step-by-step solution

Given, _ t x t^2 f(x)-x^2 f(t) t-x =1 Using L' Hospital's rule, we get _ t x t^2 f^ (x)-2 x f(t) -1 =1 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 On integrating both sides, we get f(x) x^2 =+ 1 3 x^3 +c f(x)= 1 3 x +c x^2 Also, f(1)=1 1= 1 3 1 +c(1)^2 1= 1 3 +c 2 3 =c f(x)= 1 3 x + 2 x^2 3