I am solving the following problem-
At $t=0$, a rod of length $2$ meter is maintained at $0^\circ C$. At $t>0$, the left end is insulated whereas the right end is maintained at $100^\circ C$. Now solve for the time dependent temperature equation.
Clearly, the Initial condition is $u(x,0)=0$ and the boundary conditions are $$ \left.\frac{\partial u}{\partial x}\right|_{x=0}=0$$ and $$u(2,t)=100 $$
I proceed to use separation of variables. Clearly $X=c_1\cos(kx)+c_2\sin(kx)$ and $T=e^{-\alpha^2k^2t}$. From the first boundary condition, clearly $c_2=0$. Thus, $u=X(x)T(t)=c_1\cos(kx)e^{-\alpha^2k^2t}$.
Now, I can't proceed from here and have no idea how to use the second BC.
How to go about it? Thanks in advance.