In solving the above second-order equation, it is common to rewrite it in terms of two coupled first-order equations with the velocity
$$ v(t)=\frac{dx}{dt}, $$and the acceleration
$$ \frac{dv}{dt}=F(t)-\eta v(t)-x(t). $$With the initial conditions \( v_0=v(t_0) \) and \( x_0=x(t_0) \) defined, we can integrate these equations and find their respective solutions.