I am studying chapter 10 (Partial Differential Equations and Fourier Series) of Boyce's Elementary Differential Equations and I stumbled upon this question:
Consider the conduction of heat in a rod $40~\rm cm$ in length whose ends are maintained at $0^\circ\rm C$ for all $t > 0$. In the below question find an expression for the temperature $u(x, t)$ if the initial temperature distribution in the rod is the given function. Suppose that $α_2 = 1$.
$u(x, 0) = \begin{cases} x & \ 0 \le x \lt20\\ 40-x & \ 20 \le x \le40\\ \end{cases} \\$
While I was checking the soluotion of this question I found online I saw in one of the steps involved below is treated as zero:
$$\cos\frac{\frac{(n\pi L)}{2}}{L} = 0.$$
I just wish to ask why is that the case that the above equals zero? For instance if $n = 2$ shouldn't the above equals $\cos(\pi) = -1$?
Thank you!
*this link is a screenshot of the solution I found online: !https://i.stack.imgur.com/n5SLR.jpg ; link to the full solution is here: !https://math.berkeley.edu/~ogus/old/Math_54-05/HW%20solutions/homework509.pdf
