Given that $f(x)$ and $g(x)$ are independent solutions of a linear homogeneous differential equation on $(a, b)$, which of the following must also be solutions?
D) both (A) and (B)
E) both (B) and (C)
My answer would be D, since any linear combination of independent solutions should also be a solution to the given equation, but the answer key states that only A, that is, the zero solution, is the correct one. Am I overlooking a subtlety in the wording of the question, or is D indeed the correct answer? I namely can't see why B wouldn't also be a solution.
This question is straight from a GRE preparation book, by the way, and the answer key is from UCLA's GRE preparation course website.