Solution to wave equation case three

Question

I know how to find the solution to the wave equation. When I solve it, there are three cases, when A (constant) is equal to zero, positive and negative. It is this third case, when the constant is negative which we use to continue solving the wave equation. But does anybody know why we use this third case and not the other two? Is it because we need to have boundary conditions and so we use the negative case. Thanks

Answer 1

Because only in the third case we get the nontrivial solution of the corresponding BVP. In both case $1$ and case $2$ we'll get a trivial solution. Note that on 3rd case we never equate the constant of integration to zero cause it gives a trivial solution, instead of that we equate the $\sin$ term to zero, for which we get the $n\pi$ term. That is only because we always try to avoid trivial solution.