I have worked out the solution to an ODE using two methods, with solutions as follows: $$A e^{-5t} + B e^{3t}$$ $$\text {and}$$ $$A\begin{pmatrix} \;1 \\ {-5} \end{pmatrix} e^{-5t}+B\begin{pmatrix} 1 \\ 3 \end{pmatrix} e^{3t}$$
Does the second give more information? I am not sure how to interpret the second solution. I believe that the second solution is actually two separate solutions $y$ along each line. Is that correct? Which would also make sense given that the first equation is along the top line of the second
Suppose the solution to an ODE (say $\ddot{x} + 2\dot{x} - 15x = 0$) is given by $$x(t) = Ae^{-5t} + Be^{3t}.$$ Notice that when you differentiate this you get $$\dot{x}(t) = -5Ae^{-5t} + 3Be^{3t}$$ which is the second row of your vector solution $$ \left(\begin{matrix}x\\ \dot{x}\end{matrix}\right) =A\left(\begin{matrix}\ 1\\ -5\end{matrix}\right)e^{-5t} +B\left(\begin{matrix}1\\ 3\end{matrix}\right)e^{3t}. $$ As for which gives more information; the information is there in both. I find the second form more convenient as it has all the things i need for phase plane analysis.