Quick pre-exam question..
Say we have some sort of heat equation with boundary conditions:
u(0,t) = 0,
u(L,t) = 0
where u = u(x,t)
And using separation of variables we get to;
X''/X = (1/a^2) T'/T
where X = X(x), T = T(t), a is the alpha from the heat equation.
So both sides of the equation must be constant;
(A): X''/X = k
(B): (1/a^2) T'/T = k
Now according to my textbook k must be negative, so that T goes to zero as t --> infinity.
I can't see why this is the case. What determines whether k is negative or positive? Is it determined by the boundary conditions - if the boundary conditions were derivatives (ie u'(some x, some t) = 0) would that produce a positive constant?
I'm confused because as far as I understand, a -ve constant would leads to a basis involving cosines and sines, whereas a +ve constant would lead to a basis involving hyperbolic sines/cosines. I don't know when to use which!