I'm attepmting to solve a second order partial differential equation using the seperation of variable method, but have gotten up to where i have reduced to ordinary differential equations and has been so long since i did them i'm confused.

If i have say;

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

And i try a solution of the form X =

hence X' =

and X'' =

when substituted in gives;

hence;

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) =

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.