What is going on here? (Heat equation w/ Neumann conditions)

[Jamin2112]

Homework Statement

Solve the heat equation

u_t=u_xx​

on the interval 0 < x < 1 with no-flux boundary conditions. Use the initial condition

u(x,0)=cos ∏x​

Homework Equations

We eventually get u(x,t)= B₀ + ƩB_ncos(n∏x/L)exp(-n²∏²σ²t/L²)

where

L=1 and σ=1 in our case.

B₀ is given by (1/L)∫_[0,L]u(x,0)dx

and B_n by (2/L)∫_[0,L]u(x,0)cos(n∏x/L)dx

The Attempt at a Solution

Well, I keep getting B_n = (2/L)∫_[0,L]u(x,0)cos(n∏x)dx = 0 after using the trig identity cos(u)cos(v)=(1/2)(cos(u+v)+cos(u-v)).

What's happening here? I've checked my steps many times.

 

[Jamin2112]

Thoughts?