Originally Posted by **CaptainBlack**

Now you have me slightly confused here. As I understand things a pair

$\displaystyle \{f(x),g(x)\}$ is a base solution of $\displaystyle y''+p(x)y'+q(x)y=0$

means that $\displaystyle f$ and $\displaystyle g$ are linearly independent

(have non-zero Wronskian), and $\displaystyle Af(x)+Bg(x)$ is a general

solution of $\displaystyle y''+p(x)y'+q(x)y=0$.

Your pairs both can be used to give a general solution, and with a bit

of rearrangement can be seen to give the same solutions for suitable

choices of the multipliers.

RonL