Let `f:(0,oo)vec(0,oo)` be a differentiable function satisfying, `xint_0^x(1-t)f(t)dt=int_0^x tf(t)dtx in R^+a n df(1)=1.` Determine `f(x)dot`

Explanation

We have `x int_(0)^(x)(1-t)f(t)dt=int_(0)^(x)tf(t)dx`
Differentianting both sides w.r.t `x`, we get
`x(1-x)f(x)+int_(0)^(x)(1-t)f(t)dt=xf(x)`
or `x^(2)f(x)=int_(0)^(x)(1-t)f(t)dt`
Differentiating both sides w.r.t again, we get
`x^(2)f'(x)+2xf(x)=(1-x)f(x)`
or `(f'(x))/(f(x))=(1-3x)/(x^(2))`
or `int_(f'(x))/(f(x)) dx=int_(1-3x)/(x^(2)) dx`
or `logf(x)=-1/x-3logx+logc`
or `log[(f(x))/c]=(-1)/x-3logx`
Given `f(1)=1` or `log(1/c)=1` or `c=e`
`:. log((f(x)x^(3))/e)=-1/x`
or `f(x)=1/(x^(3))e^((1-1/x))`