Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics V / 5 / b / iii

When $\alpha >n$, set $\mu =\sqrt{{\alpha }^2-n^2}>0$. The reduced ordinary differential equation is $F^{\prime \prime }-{\mu }^{2}F=0$. Enforcing the same initial conditions givesThe corresponding overdamped normal mode can also be writtenBoth rates in the exponential functions are negative because $0<\mu <\alpha$, so the hyperbolic sine factor does not prevent decay.