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I am trying to solve a partial differential equation but I am confused on one part.

The Setup:

$u_t = u_{xx}$ on 0 < x < 1, t > 0. BCs: $u_x(1,t) = u(1,t)$ & $u_x(0,t) = 0$.

I run into issues when I perform separation of variables.

For the case $\lambda = 0$ I get a trivial solution.

For the case $\lambda > 0$, I let $\eta = \lambda^2$ and using the BCs get $\eta = -cot(\eta)$.

For the case $\lambda < 0$, I let $\eta = -\lambda^2$ and using the BCs get $\eta = coth(\eta)$.

I have looked at the graphs for both of these cases and I don't understand how I am supposed to decide the value of $\lambda$. Usually the problems have something that indicates a non-trivial solution is only possible for one case. Looking at the graphs of each case I don't see how one of the two indicates a trivial solution.

Usually we find a set of $\lambda$ values so I think that the periodic nature of cot($\eta$) would be the non-trivial case. However, this problem makes me question whether or not I understand what the cases are supposed to tell me about the original PDE.

djblue
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    It can happen that you end up with 1 or 2 solutions that are negative, depending on the endpoint conditions. For example $e^{-x}$ satisfies $X(0)+X'(0)=0$ and $X(1)+X'(1)=0$. However, in your case, this does not happen. $\coth(\eta)=\frac{e^{\eta}+e^{-\eta}}{e^{\eta}-e^{-\eta}}=1+\frac{2e^{-\eta}}{e^{\eta}-e^{-\eta}}=1+\frac{2}{e^{2\eta}-1}$ – Disintegrating By Parts Oct 02 '15 at 08:36

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