This is very similar to the Method of Undetermined Coefficients encountered in normal differential equations, with some slight exceptions to the "rules" of guessing. Actually, there's only one rule extra. In the normal method of undetermined coefficients when there was a conflict of the characteristic equation with the particular solution, there was a multiplication by the independent variable. In this case we multiply by $\mathbf {A} t+\mathbf {B}$ to include more possible solutions. That, and when working through the problem thoughts about the signifigance of getting a trivial solution when finding eigenvalues must be kept in mind. Other than that, it's pretty much as it was, and is a very powerful method (although it can be quite tedious).