Consider the Langevin equation
$$
\frac{d v}{d t}=-\gamma v+\frac{1}{m} F^{\prime}(t)
$$
where the first term on the right is a phenomenological expression for the slowly varying part of the interaction force, whose rapidly fluctuating part is denoted by $F^y(t)$ If $F^{\prime}$ is neglected, the solution of the resulting equation is $v=u \exp (-\gamma t)$ where $u$ is a constant. In the general case where $F^{\prime} \neq 0$, assume a solution of the same form with $u=u(t)$ and ehow that the solution of the Langevin equation gives for the velocity at time $t$ the result
$$
v=v_0 e^{-\gamma^t}+\frac{1}{m} e^{-\gamma^1} \int_0^t e^{\gamma^{\prime \prime}} F^{\prime}\left(t^{\prime}\right) d t^{\prime}
$$
where $v_0=v(\mathbf{0})$.