Lets take the equation $\Delta u=0$, where $u=u(x,y)$, $\Delta $ is the laplacian in two dimensions (x and y), and $0<x<L, 0<y<L$. Lets assume homogenous Dirichlet boundary conditions in the x-direction. If we use seperation of variables, then we get $\frac{X(x)''}{X(x)}=-\frac{Y(y)''}{Y(y)}=-k^2$. Here is where I usually get into trouble: If we use the above standing seperation of variables then, we get $X_n(x)=\sin(k_nx), k_n=\frac{n\pi}{L}$. But we could just as easilly have seperated the variables so that we had gotten $-\frac{X(x)''}{X(x)}=\frac{Y(y)''}{Y(y)}=-k^2$, and then we would have gotten $X_n(x)=\sinh(k_nx)$ with some $k_n$.
My question is this: are those solutions equivallent, or not?
Thanks!