If i have say;

X'' - X/(k^2) = 0

And i try a solution of the form X =\(\displaystyle e^{(\lambda x)}\)

hence X' = \(\displaystyle \lambda e^{(\lambda x)} \)

and X'' = \(\displaystyle \lambda^2 e^{(\lambda x)} \)

when substituted in gives;

\(\displaystyle \lambda^2 e^{(\lambda x)} - \frac{e^{(\lambda x)}}{k^2} = 0\)

hence; \(\displaystyle \lambda = + (1/k) or - 1/k \)

is this correct up to here? and the following is where i'm confused,

this seems to me to be real distinct solutions hence general solution should be;

X(x) = \(\displaystyle A.e^{\frac{x}{k}} +B.e^{\frac{-x}{k}} \)

but i remember the lecturer saying that this should be a trig or hyperbolic solution, as there is no complex roots i'm gussing hyperbolics?

Any help would be greatly appreciated, i'm very hazy on this stuff,

Cheers.