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

When $\alpha =n$, the reduced ordinary differential equation is $F^{\prime \prime }=0$. The initial conditions $F(0)=0$, $F^{\prime }(0)=1$ giveThis is the critically damped normal mode, also obtained by taking $\nu \to 0$ in $\sin (\nu t)/\nu$.