# Given independent solutions $f(x)$ and $g(x)$ of a linear homogeneous DE on $(a,b)$, which must also be solutions?

Given that $f(x)$ and $g(x)$ are independent solutions of a linear homogeneous diff erential equation on $(a, b)$, which of the following must also be solutions?

A) 0

B) $2f(x)-3g(x)$

C) $f(x)g(x)$

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.

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I agree with you, the way the question is worded. – user84413 Aug 19 '13 at 0:30
@user84413, thanks, and it seems by the comment upvotes, a lot of people would agree with you, which I'm glad to see. I was unsure of whether the interval condition or the wording of "independent" rather than "linearly independent" would change anything, but couldn't come up with a reason as to why that would be the case. – Ryker Aug 19 '13 at 2:37

Consider this example:

$x_1 + 3x_2 = 0$

$2x_1 + 6x_2 = 0$

It has the trivial solution $(0, 0)$ but also the non-trivial solution $(-3, 1).$

There are infinitely many other linear combinations that are solutions: e.g. $(-3n, 1n)$ for $n > 0$.

Could it be that the linear combinations for a homogeneous DifEQ are not considered independent, thus why $B$ is not an answer?

(Reference: 2nd example on page 2 http://www.math.hawaii.edu/~lee/linear/sys-eq.pdf)

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I don't know, but the wording of the question says f and g are linearly independent. So I still don't see where, unless it's an honest mistake on their part, they get that the function in B is not a solution. – Ryker Sep 14 '13 at 1:09