How can I solve $$f'(x-1)-f''(x)=g(x),$$ or similar delay differential equations (DDEs) where there's some function on the r.h.s.? Is there a general method.
I tried the following: Suppose $f(x)=Ce^{ax}$. Then $f'(x)=Cae^{ax}$ and $f''(x)=Ca^2e^{ax}$, and $$Cae^{ax-a}-Ca^2e^{ax}=g(x).$$ Then I'm stuck. Ordinarily we could continue if $g(x)\equiv 0$, but this is not necessarily the case. I'm wondering if I can use a Laplace transform, e.g. $$C\int_0^\infty ae^{-a}-Ca^2dx=\int_0^\infty g(x)e^{-ax}dx,$$ but not sure if this is the right way. I believe this approach is to do with the characteristic polynomial method, but I have also heard of the "method of steps" too.
I think I'm on the wrong footing. Maybe I need to assume $f(x)=Ce^{ah(g(x))}$...
Any help, references, hints, or guidance welcomed.