Ytukyg

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 $gamma = 7$.

$$ ddot{theta} + gammadot{theta} + sin(theta) = 0 tag{1} $$

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$$ddottheta+7dot theta+theta=0$$by defining state variables $theta_1=theta$ and $theta_2=dottheta$ we have$$begin{bmatrix}dottheta_1\dottheta_2end{bmatrix}=begin{bmatrix}0&1\-1&-7end{bmatrix}begin{bmatrix}theta_1\theta_2end{bmatrix}$$If we try to express the main problem we have$$begin{bmatrix}dottheta_1\dottheta_2end{bmatrix}=begin{bmatrix}0&1\-{sintheta_1over theta_1}&-7end{bmatrix}begin{bmatrix}theta_1\theta_2end{bmatrix}$$