Need help to the following question. Is there a way to express the following differential equation into a matrix form? Since there is a $\sin(\theta)$ term in the equation, how to handle this? Thanks in advance! Please consider $\gamma = 7$.
$$ \ddot{\theta} + \gamma\dot{\theta} + \sin(\theta) = 0 \tag{1} $$
given the conditions
$$ \theta(0) = \theta_{0} \\ \dot{\theta}(0) = \dot{\theta}_{0} \tag{2} $$
A matrix form, induces a linear matter inside the problem which is inapplicable to here since existence of $\sin(.)$ term makes the problem generally non-linear regardless of initial conditions. Therefore a matrix form can be attained thereby tolerating errors. If we approximate $\sin x$ with $x$ for small enough $x$ we obtain$$\ddot\theta+7\dot \theta+\theta=0$$by defining state variables $\theta_1=\theta$ and $\theta_2=\dot\theta$ we have$$\begin{bmatrix}\dot\theta_1\\\dot\theta_2\end{bmatrix}=\begin{bmatrix}0&1\\-1&-7\end{bmatrix}\begin{bmatrix}\theta_1\\\theta_2\end{bmatrix}$$If we try to express the main problem we have$$\begin{bmatrix}\dot\theta_1\\\dot\theta_2\end{bmatrix}=\begin{bmatrix}0&1\\-{\sin\theta_1\over \theta_1}&-7\end{bmatrix}\begin{bmatrix}\theta_1\\\theta_2\end{bmatrix}$$