Integral equation: $x f(x) = \int _0 ^x \int _0 ^t f(u) \ \Bbb d u \ \Bbb d t$

Question

Would you please find the function $f$ such that $$x f(x) = \int _0 ^x \int _0 ^t f(u) \ \Bbb d u \ \Bbb d t \quad ?$$ Thank you.

Answer 1

We have: $$xf(x)=\int_0^x\int_0^tf(u)\mathrm du\mathrm dt$$

Differentiating both sides w.r.t. $x$: $$f(x)+xf'(x)=\int_0^xf(u)\mathrm du$$

Differentiating both sides w.r.t. $x$: $$f'(x)+f'(x)+xf''(x)=f(x)$$

Using $y$ to substitute $f(x)$: $$xy''+2y'-y=0$$