1549380323-Statistical Mechanics Theory and Molecular Simulation

674 Laplace transforms

̃x(s) =

1

s^2

+a^2

̃x(s)

s^2

. (D.19)

Solving this for ̃x(s), we obtain

̃x(s) =

1

s^2 −a^2

, (D.20)

and, as we have already shown, the inverse Laplace transform of ̃x(s), which gives us
the solutionx(t) is

x(t) =

1

a

sinh(at). (D.21)