# differential equation system order

my question is very easy: I started study dynamic systems and I have this system:

$Mz_1''(t) +\beta z_1'(t) = F(t)$

$Mz_2''(t) - \beta z_2'(t)+kz_2(t)=\beta z_1'(t)$

where $M, \beta, k$ are constants and ' is the order of diff. equation.

Now I have to choose the state variables, that they are as the order of the system. Which is the order of this system? 4 or 2? So how much state variables can I choose?

Regards

• solve the first equation and substitute it in the second equation and then solve the second equation Oct 17, 2015 at 20:20
• I tryed to find $z_1'$ and substitute it in the second equation, but I still have $z_1''$ . Oct 17, 2015 at 20:36
• Do you know how to get the complementary solution and the particular solution? Oct 17, 2015 at 20:39
• I don't solve the system I have to choose the order of the system and then choose the state variables and write i / s / o rappresentation of the mechanical system Oct 17, 2015 at 20:41

$$x_1 = z_1, x_2 = z_1', x_3 = z_2, x_4 = z_2'$$
$$\begin{bmatrix}x_1' \\ x_2' \\ x_3' \\ x_4'\end{bmatrix} = \begin{bmatrix}* & * & * & * \\ * & * & * & * \\ * & * & * & * \\ * & * & * & * \end{bmatrix} \begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4\end{bmatrix} + \begin{bmatrix}* \\ * \\ * \\ *\end{bmatrix}$$