My question is about finding a particular integral for a second order inhomogeneous differential equation (constant coefficients) when the inhomogeneous term is of the same form as the complementary solution to the homogeneous equation. It is clear that multiplying the complementary solution by the independent variable produces a function, Y, that is not a solution to the homogeneous equation. Should I, however, always expect Y to satisfy the inhomogeneous equation? Or is Y merely a good guess for most basic undergraduate problems that I am likely to encounter?
Thank you in anticipation of your help.