Here I have a system of nonlinear differential equations:
$ (M+2m)\ddot{x} + m(l_1 \ddot{\theta}_1\cos\theta_1 - l_1\dot{\theta}_1^2\sin\theta_1) + m(l_2\ddot{\theta}_2\cos\theta_2-l_2\dot{\theta}_2^2\sin\theta_2) = F $
$ l_1\ddot{\theta}_1 + \ddot{x}\cos\theta_1 - g\sin\theta_1 = 0 $
$ l_2\ddot{\theta}_2 + \ddot{x}\cos\theta_2 - g\sin\theta_2 = 0 $
States are defined by me like this:
$x_1 = \theta_1 , x_2 = \dot{x_1} = \dot{\theta}_1,x_3 = \theta_2, x_4 = \dot{x_3} = \dot{\theta}_2,x_5 = x, x_6 = \dot{x_5} = \dot{x}$
I mean actually what I need to do is to linearize this system about all the states are equal to $0$. But I cannot find the $\cdots$ places below
$ \dot{x_1} = x_2 \\ \dot{x_2} = \cdots \\ \dot{x_3} = x_4 \\ \dot{x_4} = \cdots \\ \dot{x_5} = x_6 \\ \dot{x_6} = \cdots \\ $
How can I find $\dot{x_2},\dot{x_4},\dot{x_6}$?
Thanks