11.5.2.14 Fourier method

Problem: Find the solution of the Cauchy problem for the heat equation: \[ \begin{array}{l} u_{t}=8 u_{x x}, \quad u(x, 0)=5 \cos 2 \pi x+6 \cos 3 \pi x, \\ u_{x}(0, t)=u_{x}(6, t)=0, \quad a=\sqrt{8}=2 \sqrt{2}, \quad l=6 . \end{array} \]