Insulated ends $u_x=0$ become Neumann conditions on $X$

Consider the boundary value problem $u_t=u_{xx}$, $0<x<c$, $t>0$, with $u_x(0,t)=0=u_x(c,t)$, $t>0$, and $u(x,0)=f(x)$. Substituting $u(x,t)=X(x)T(t)$ we generate the following Sturm–Liouville problem for $X(x)$: