Consider a linear system governed by the differential equation
m\ddot{x}(t)+c\dot{x}(t)+kx(t)=f(t)The accompanying sketch shows one physical system that is governed by this differential equation, with m being a mass, k being a spring stiffness, and c being a dashpot value. This system is called the single-degree-of-freedom system and will play a key role in our study of stochastic dynamics, but at this point it is simply one more example of a relatively simple dynamical system. Find the impulse function such that Eq. 5.2
x(t)=\int_{-\infty }^{\infty }{f(s)h_{x}(t-s)ds} \equiv \int_{-\infty }^{\infty }{f(t-r)h_{x}(r)dr}describes the solution of the problem.