Solving linear non-homogeneous integral equation Is it possible to solve equations of the kind:
$$x(t) = \int \limits_t^T c_1  x(s) ds +c_2 $$ with $c_1$ and $c_2$ being some constants and if I know $x(0)$?
Or do I need more assumptions on $x$?

Hint:
what about differentiating both sides and solving for $x(t)$ (provided $x(t)$ is differentiable - thanks to @LiuGang for pointing out)? You will need to recall that: 

$$ \color{blue}{\frac{\mathrm{d} }{\mathrm{d} t} \int^t_{a} f(s)  \, \mathrm{d}s =f(t), \quad a \in \mathbb{R}}$$

and, of course, that:

$$\color{blue}{ x'(t) + c \, x = 0 \implies x(t) = x_0 e^{-c  t}, \quad x_0 = x(t=0)}$$

Hope this helps.
Cheers!

Question: will there be any relation between $c_2$ and $T$? 
