Two first-order differential equations

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.

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