Let `f:RtoR` be a differntiable function satisfying `f(x)=x^(2)+3int_(0)^(x)e^(-t^(3)).f(x-t^(3))dt`. Then find `f(x)`.

Explanation

`f(x)=x^(2)+3int_(0)^(x^(1/3))e^(-t^(2))t^(2)(x-t^(3))dt`……………1
Put `x-t^(3)=z` in second term
`:.f(x)=x^(2)+int_(0)^(x)e^(z-x)f(z)dz`
`=x^(2)+e^(-x)int_(0)^(x)e^(z)f(z)dz`
Defferentiating w.r.t `x` get
`f'(x)=2x+e^(-x)e^(x)f(x)-e^(-x)int_(0)^(x)e^(z)f(z)dz`
`:.f'(x)=2x+x^(2)`
`:.f(x)=(x^(3))/3+x^(2)+c`
From 1 `f(0)=0`
`f(x)=(x^(3))/3+x^(2)`