Solution Integro differential equation

Question

I try to solve the two integro differential equations

$f(at)=\frac{df(t)}{dt}$ and $f(at)=\int_{0}^{t}f(\tau)d\tau$ $a\gt0$.

Do you have an idea to suggest to me.

Thank you very much for your kind help.

Answer 1

Hint: $$\Longrightarrow \frac{df(t)}{dt} = \int_0^t f(\tau)d\tau$$ Define $F(t):=\int_0^t f(\tau)d\tau$, then $$F''(t) =F(t)\tag{1}$$ the solution of $(1)$ is $F(t) = A\cdot e^t+B\cdot e^{-t})$

Then you deduce $f(t)$