Finding the particular/complementary solution from a laplace transform

[eggshell]

Say you find the laplace transform V(s) and want to switch it back to the time domain, once you've done this, how do you determine which parts of the total solution correspond to the complementary solution and particular solution respectively? Do you just find which parts approach zero as time increases to infinity, and label that as the complementary, or is there more to it than that?

 

[HallsofIvy]

Strictly speaking, you can't. If, for example, y(x)= Cf(x)+ Dg(x)+ h(x) is a solution to the differential equation, where C and D are undetermined constants, so that Cf(x)+ Dg(x) is the "complimentary solution" and h(x) is the "particular solution, we could just as easily write y(x)= (C- 1)f(x)+ (D- 2)g(x)+ (f(x)+ 2g(x)+ h(x)) so that (C- 1)f(x)+ (D- 2)g(x) is the "complimentary solution" and f(x)+ 2g(x)+ h(x) is the "particular solution". In other words, what part of a solution is "complimentary" and which is "particular" is purely a matter of choice.