An integral equation $x(t)=a-\left(1-x(0)\right)e^{-\int_{t_1=0}^tx(t_1)dt_1}$

Question

Consider ${\rm x}\left(0\right)$ is a fix positive real number, and we have following equation:

$$ {\rm x}\left(t\right) =a - \left[1 - {\rm x}\left(0\right)\right]\exp\left(-\int_{0}^{t}{\rm x}\left(t_{1}\right)\,{\rm d}t_{1}\right) $$

Is there any way to solve the above equation ?. Do they always have solution ?.

Any help or reference is appreciated.

Answer 1

Note that

$$x'(t) = [1-x(0)] x(t) \, e^{-\int_0^t dt' \, x(t')} = x(t) [a-x(t)]$$