Find recurrent formula for differential equation

Question

Here is the problem: $$F'(u)=\frac{l}{c}[F(u)-F(u-m)]$$

$$F(u)=\frac{c-lm}{c}e^\frac{lu}{c} $$ for $0\le u \le m$

Prove that for $km\le u \le (k+1)m$

$$F(u)=F(km)e^\frac{l(u-km)}{c}-\frac{l}{c}\int_{0}^{u-km} F(u-y-m)e^\frac{ly}{c}.$$