Hi... I've been set this question as homework, and I'm really struggling with it - I'd really appreciate any pointers that anyone could give me - I'm NOT asking for the answer, but instead would appreciate any methodological hints.

It's about a bar adjusting to ambient temperature.

The equation is:

\frac{\partial y}{\partial t} =\kappa \frac{\partial^2 y}{\partial x^2}, t > 0, x \in[0, \pi] with boundary conditions:

y(0,t) = T_0
\frac{\partial y}{\partial x}(\pi,t) - \kappa y(\pi,t)=-\kappa T_0

T_0 is just a constant, but using the substitution v(x,t) =  y(x,t) - T_0, I've gotten:

\frac{\partial v}{\partial t} =\kappa \frac{\partial^2 v}{\partial x^2}, t > 0, x \in[0, \pi]

with boundary conditions:
v(0,t) = 0 (first part of the Dirichlet conditions)

\frac{\partial v}{\partial x}(\pi,t) = \kappa v(\pi,t)

I'm also going to make the ansatz that v(x,t) = X(x)T(t), implying:

\lambda = \frac{\partial^2 X}{\partial x^2} = \frac{1}{\kappa} \frac{\partial T}{\partial t}

I'm aware that there are obviously differing cases of \kappa to deal with (positive, negative, and zero), but I'm stuck when dealing with the boundary conditions - I'm assuming I have to somehow show that the Dirichlet conditions hold for each case (or that it is trivial), but I can't see how to use \frac{\partial v}{\partial x}(\pi,t) - \kappa v(\pi,t)=0 to show that v(\pi,t) = 0 for each case of \kappa.

Does anybody have any ideas? I just can't see how to use this information to solve the equations - without proper boundary conditions, I can't solve anything at all, and I've been at this for a few hours now.