vector field for 2nd order differential eq. This is a real exam question I was not able to solve:
Draw the vector field corrsponding to the differential equation:
$m\ddot{x} = -\omega^2x + \gamma \dot{x} + f(t)$  
What's so odd about this is that this is a 2nd order differential equation. Any ideas on this?

An $n^{th}$ order diﬀerential equation can be converted into an $n-$dimensional system  of first order  differential equations.
We have:


*

*$x_1 = x$

*$x'_1 = x' = x_2$

*$x'_2 = x'' = \dfrac{1}{m} (-\omega^2x + \gamma~ x' + f(t)) = \dfrac{1}{m} (-\omega^2 x_1 + \gamma~ x_2 + f(t))$


Our reduced system is:
$$\begin{aligned}
x'_1 & = x_2 \\
x'_2 & = \dfrac{1}{m} (-\omega^2~ x_1 + \gamma~ x_2 + f(t)) 
\end{aligned}$$
