Seperation constant giving a harmonic dependence. (Seperation of variables)

xago

1. The problem statement, all variables and given/known data


3. The attempt at a solution
I'm on part b) where it asks which seperation constation gives a harmonic time dependence. From part a) I deduced the equation [itex]\frac{d^{2}T}{dt^{2}}[/itex][itex]\frac{1}{T}[/itex] = a constant. I'm choosing the constant [itex]k^{2}[/itex] and my question is does it matter if the constant is negative or positive? I have seen in textbooks that a positive constant gives the solution T(t) = Aexp(-kt) + Bexp(kt) whereas a negative one would be Acos(kt) + B sin(kt). Are both solutions equivalent or does only one of them give a harmonic time dependence (My guess would be the sin/cos one is the proper answer for this question.)

tiny-tim

hi xago!
yes, you're right … harmonic has to be periodic, and only sin/cos will be periodic

(exp will be either runaway or decay )