How do I know if I am using the right particular integral when solving a differential equation

Particular integrals will generally be derived in two different ways, depending on the type of the differential equation that you are solving. For example if it was a linear first order ODE, then you would have to solve e^(integral P.dx). Or, if you had a second order linear ODE, then you have have to guess the answer, (depending on the forcing function in the inhomogenous case), substitute the guess and its higher order derivatives into the ODE and solve for the unkown co-efficicient(s). The question becomes (" is this the right one " ) and the best way to find out is simply through an example. If your particular integral already appears in the complementary function you have just derived, then determining the unknown co-efficient will be impossible as it will lead to solving an inconsistent set of equations. in that scenario you would have to multiply by x (or even by x² if that appears again). In the case of a first order ODE. The particular integral should be chosen such that the ODE simplifies into a derivative of a product of y and the integrating factor. Getting the right one only comes through practice and experience.