Rewrite the solution to Problem 2 and identify the initial value Green's function.

Data from Problem 2

Find product solutions, \(u(x, t)=b(t) \phi(x)\), to the heat equation satisfying the boundary conditions \(u_{x}(0, t)=0\) and \(u(L, t)=0\). Use these solutions to find a general solution of the heat equation satisfying these boundary conditions.