Question

how do you verify a fundamental solution set to a nonhomogeneous equation

Answer 1

Your fundamental set of solutions are the solutions to the homogeneous equation that corresponds to your non homogeneous equation, right? Your entire solution set is made up of the homogenous part and the particular part.

Anyway, to check if your fundamental solutions are a solution, you look at the Wronskian. For example, suppose f(x) and g(x) are two fundamental solutions. Then, evaluate:

| f g |
| f' g' |

If the determinant of that matrix is nonzero, then f+g is your homogenous solution. If it is zero, then f and g are linearly dependent and as a result, either of them are sufficient for your homogeneous solution.