Consider the one dimensional wave equation $$ u_{tt}=c^2u_{xx} $$ where the initial conditions are $$ \begin{cases} u(x,0) = Ax\\ u_x(0,t)=u_x(L,t)=0. \end{cases} $$ With separation of variables $u(x,t)=X(x)T(t)$, I obtain:
$$X=B\cos([\sqrt{\lambda}]x)$$
and
$$-[\sqrt{\lambda}]B\sin([\sqrt{\lambda}]L)= 0.$$
We get $$[\sqrt{\lambda}]=2n\pi/L$$
However, the answer states that it has to be $(2n-1)$ instead of $2n$ which suggests that a $-1$ value for the cosine value is required for the given situation.
I'm wondering what I've missed to not get the correct value.
Thank you for your time.
I do not know how to use mathjax so I will state the answer below in image format:
answer Thanks again.
