Question

Let `f:R -(0,oo) ` be a real valued function satisfying `int_0^1 tf(x-t) dt =e^(2x)-1` then which of the following is/are correct?

Answer 1

`int_(0)^(x) tf(x-t)dt=e^(2x)-1`
`impliesint_(0)^(x)(x-t)f(t)dt=e^(2x)-1`
`impliesx int_(0)^(x)f(t)dt-int_(0)^(x)tf(t)dt=e^(2x)-1`
Differentiating both sides w.r.t `x`, we get
`xf(x)+int_(0)^(x)f(t)dt-xf(x)=2e^(2x)`
`impliesint_(0)^(x)f(t)dt=2e^(2x)`
`impliesf(x)=4e^(2x)`