I am interested in solving the following equation: $$ W'(s) = \big( W(s) - W(s-1) \big)\cdot \lambda$$ where $W(s)=e^s$ for $s \leq 1.$
This might be quite simple yet when I searched for 'linear differential-difference equations' and for 'mixed differential-difference equations' I was able to find specific articles dealing with methods for general differential equations. I have found an article studying approximate solutions for these yet aren't there elementary methods for finding solutions explicitly?
Any references are appreciated.