 Solution

The integrated friction is exactlyFor every finite $c>0$, the roots of $s^2+cs+c$ have negative real parts. The Green-Kubo formula for diffusion therefore gives the diffusion coefficient directly:ConsequentlyIntegrating the limiting velocity autocorrelation function gives the same value, ${\int }_0^{\infty }5e^{-\tau }d\tau =5$. Computing $D_c$ before taking the limit also avoids having to justify an interchange of the infinite-time integral and the parameter limit.