The Problem
I have a very large system of coupled ODEs which is similar to:
$$\begin{multline} \begin{bmatrix}m_1 & 0 \\ 0 & m_2\end{bmatrix} \begin{bmatrix}\ddot{x}_1 \\ \ddot{x}_2\end{bmatrix} +\begin{bmatrix}c_1+c_2 & -c_2 \\ -c_2 & c_2+c_3\end{bmatrix} \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2\end{bmatrix} +\begin{bmatrix}k_1+k_2 & -k_2 \\ -k_2 & k_2+k_3\end{bmatrix} \begin{bmatrix} x_1 \\ x_2\end{bmatrix} \\+\begin{bmatrix}g_1 & 0 \\ 0 & g_2\end{bmatrix} \begin{bmatrix} \sin(\omega t -\phi_1) \\ \sin(\omega t -\phi_2)\end{bmatrix} =\mathbf{0} \end{multline}$$
where in a more compact matrix notation the above becomes:
$$M\ddot{X}+C\dot{X}+KX+G\sin(\omega t - \Phi)=\mathbf{0} \tag{1}$$
Using this equation, I would like to find the sensitivity of variable $\dot{x}_1$ with respect to $\dot{x}_2$, i.e. $\dfrac{\partial \dot{x}_1}{\partial \dot{x}_2}$.
My Attempt
My fist task is to set equation (1) to first order format by letting $\mathbf{X}=[X,\dot{X}]^T$:
$$\mathbf{\dot{X}}=\begin{bmatrix} \mathbf{0} & \mathbf{I} \\ -M^{-1}K & -M^{-1}C \end{bmatrix}\mathbf{X}+\begin{bmatrix} \mathbf{0} \\ G\sin(\omega t - \Phi) \end{bmatrix} \tag{2}$$ $$ \Rightarrow \dot{\mathbf{X}} = \mathbf{F}(t,\mathbf{X}) \tag{3}$$
Then I take the derivative of $\mathbf{\dot{X}}$ with respect to $\dot{x}_2$, such that:
$$\frac{\partial \dot{\mathbf{X}}}{\partial \dot{x}_2} = \frac{\partial\mathbf{F}}{\partial\mathbf{X}}\frac{\partial\mathbf{X}}{\partial \dot{x}_2} \tag{4}$$
Where $\frac{\partial\mathbf{F}}{\partial\mathbf{X}} = \begin{bmatrix} \mathbf{0} & \mathbf{I} \\ -M^{-1}K & -M^{-1}C \end{bmatrix}$, i.e. the Jacobian of equation (2).
Thus, equation (4) is another differential equation to be solved for $\frac{\partial \mathbf{X}}{\partial \dot{x}_2}$. Since $\frac{\partial\mathbf{X}}{\partial \dot{x}_2} = \frac{\partial}{\partial \dot{x}_2}[x_1,x_2,\dot{x}_1,\dot{x}_2]^T$, I can simply extract $\frac{\partial \dot{x}_1}{\partial \dot{x}_2}$ from that vector. Initially, this approach seems reasonable to me, however in an effort to double-check the logic behind this, I realised that $\frac{\partial\dot{x}_2}{\partial\dot{x}_2}\neq1$, which should not be the case.
As a matter of fact, from this article's equation 6, I found that my equation (4) is actually used to find the sensitivity of the variables with respect to their initial conditions. So could anyone explain to me how to find $\frac{\partial\dot{x}_1}{\partial\dot{x}_2}$?
Thanks for your help in advance.