Hey, each of those have right members which are particular solutions to some homogeneous differential equation. For example, the first one has a right member which is a solution to the equation:
So the operator becomes an "annihilation" operator that we can apply to both side of the equation to convert it to a homogeneous equation:
This is the method of undetermined coefficients. Are you familiar with that method? Try it first on some simple ones. Any DE book should have a section on this subject.