I must say I'm utterly confused with the Annihilator method for solving Non-Homogeneous Constant Coefficient Second order O.D.Es. I guess it'd be better to list out my questions:- 1. Is it possible to find an annihilator for every single function out there? I mean, is it always possible to make the RHS 0 regardless of the function? 2. Is it always possible to factorise the LHS after multiplication with the annihilator? The problems I've worked with, all had neat solutions, but I'm somehow not convinced. What if the annihilator is so complex (not the mathematical sense) that it becomes incredibly hard to factorise the LHS? 3. I've sticked with this method for the reasons :- 1) that it requires a minimal memorisation of formulae (and it somehow feels more rewarding!) & 2) it is(as far as I've felt) somewhat general in application. Are there other general methods that work for a large class of functions, other than the Method of Variation of Parameters or the Method of Undetermined Coefficients? 4. Lastly, why does the Annihilator method work? Can someone please explain the theory behind this to me? I've solved so many problems by this method, yet I don't think I've understood the theory completely. I'd be really grateful if someone could explain it on a step-by-step basis? Thank you for taking the time to read this. Any and all help is appreciated.