Consider the second-order linear PDE
$$
y\_t(x,t) = y\_{xx}(x,t) - a^2y(x,t)
$$
where $a > 0$ in all cases and the equation is restricted to the domain $x = [0, X]$. If we have some way of expressing $y(x,t)$ as e.g.
$$
y(x,t) = f(x)g(t)
$$
where both $f(x)$ and $g(t)$ are known, and given boundary conditions \begin{align}
y(0, t) &= y\_0(t) \\
y\_t(0,t) &= y'\_0(t) \\
y\_x(X, t) &= 0
\end{align}

**is there an analytical or approximate solution to the PDE?**