Let's say I assumed that the answer to a PDE was U(x,t)= XT, where X,T are functions. I then further my answer by getting to a point for T'/T=kX''/X, where k is some constant given in the boundary conditions. I then continue by working on either side to find each function. Suppose I work on the right hand side (RHS). Since i know both LHS and RHS are indepent of each other, I can continue and say that RHS= -$. Now, I check the first case there $<0. So i say that $=-h^2 and get kX'' + (h^2)X=0. I am stuck here. Does X then turn out to be a function of cosM and sinM, when $<0? (M being theta, or some angle). Furthermore, X=CcosM + DsinM...right? note: I didnt include that boundary conditions as I doubt that they are needed for my question.