I encountered a fun formula that says $$e^{a \frac{d}{dx}}f(x)=f(x+a).$$ Now, I remember reading somewhere that this can be "used to solve delay differential equations". Take the simplest case, where $$x'(t)=x(t-1).$$
If you apply that exponential to both sides,
$$e^{ \frac{d}{dx}}x'(t)=x(t)$$
then it does look kind of close to a Laplace transform before integrating, so I can see the potential. But then, how do you actually solve for a solution with this?
