1. The problem statement, all variables and given/known data Find a formal solution of the heat equation u_t=u_xx subject to the following: u(0,t)=0 u_x(∏,t)=0 u(x,0)=f(x) for 0≤x≤∏ and t≥0 2. Relevant equations u(x,t)=X(x)T(t) 3. The attempt at a solution I first began with a separation of variables. T'(t)=λT(t) T(t) = Ce^(λt). X''(x)=λX(x) The boundary conditions show that X(0)=0 and X'(∏)=0. There are three general cases for the solution X(x). Case 1: When λ=0, I get X(x)=Ax+B, and using the boundary conditions get A=B=0. So the solution is a trivial solution (not what I need). Case 2: When λ=μ^2 >0, then X(x)=Acosh(μx)+Bsinh(μx) and X'(x)=Aμsinh(μx)+Bμcosh(μx). Again, using the boundary conditions lead to C=D=0, leading to the trivial solution X(x)=0. Case 3: When λ=-μ^2 <0, then X(x)=Acos(μx)+Bsin(μx). X'(x)=-Aμsin(μx)+Bμcos(μx). This is where I'm stuck. After solving I get A=0 and B≠0, as cos(μ∏)=0. How do I continue? The solution is along the lines of f(x)=u(x,0)= Summation of n=1 to infinity of (Bn*sin(nx)), but clearly that is wrong. Help? EDIT: You don't have to show that the fourier series obtained converges to the solution. After looking at it some more, I think that X(x)=sin(μx). λ_n=-(μ_n)^2=-n^2/2 (where n is a positive integer). Would that be right?