Actually I know it's going to be the sum of two exponentials with different multipliation contants and one will have the negative argument of the other.
Since, the derivative is respect to x, why not try [itex]kx[/itex] where [itex]k[/itex] is a constant (it may be complex) for your argument? In fact, the general solution has two terms [itex]y(x)=Ae^{kx}+Be^{-kx}[/itex]. When you plug this into your DE, what do you get?
so it is a complex argument as I was expecting because a complex exp can be written in terms of sines and cosines, whose 2nd derivative is their own negative.
This is a ay''+by'+cy=0 problem which has been discussed to death on every DE book. One should be able to write down the results (when b^2-4ac>0,<0 and = 0) while sleeping.