Two coupled second order differential equations I have two coupled equations in the form:
$$f''(x) + g'(x) + f(x) = 0$$
$$g''(x) + f'(x) + g(x) = 0$$
Looking at the form, i can guess a relation of the form $g(x) = \lambda f(x)$.
where $\lambda$ is some constant. I can find the constant by replacing $g(x)$ in the above equations and comparing the coefficients of every derivative. Finally i'm left with a single equation which is easily solvable. 
The question is: is this procedure legal? and is the solution that i get is the most general one?
Thanks

Hint
Make a substitute
$$
f' = p \\
g' = q
$$
then you get a system
\begin{align}
f' &= p \\
g' &= q \\
p' &= -f - q \\
q' &= -g - p
\end{align}
or in matrix form
$$
\left[ \begin{array}{c}
f \\ g \\ p \\ q
\end{array}\right ]' = \left [ \begin{array}{cccc}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
-1 & 0 & 0 & -1 \\
0 & -1 & -1 & 0
\end{array}\right ] \cdot 
\left[ \begin{array}{c}
f \\ g \\ p \\ q
\end{array}\right ]
$$
which is in form of
$$
\mathbf y' = \mathbf A \cdot \mathbf y
$$
Can you take it from here?
