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

[xago]

Homework Statement

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The Attempt at a Solution

I'm on part b) where it asks which separation constation gives a harmonic time dependence. From part a) I deduced the equation $\frac{d^{2}T}{dt^{2}}$$\frac{1}{T}$ = a constant. I'm choosing the constant $k^{2}$ 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!

… 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.)

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

(exp will be either runaway or decay )