[Solved/PDE-01004] Partial Differentaial Equations

\( \newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2}}\)

Question:-

Separate the variables and obtain most general form of the solution of the heat equation  \[\pp[u(x,t)]{t}= - \frac{1}{k} \PP[u]{x},\]   where \(k\) is a constant, satisfying the boundary conditions    \begin{eqnarray}\nonumber      \pp[u(x,t)]{x} \Big|_{x=-L}= \pp[u(x,t)]{x}\Big|_{x=L} =0, \qquad t>0.
   \end{eqnarray}   Find the solution satisfying the initial condition   \begin{equation*}
   u(x,0) = \cos\Big(\frac{3\pi x}{2L}\Big) \cos\Big(\frac{7\pi x}{L}\Big).    \end{equation*}