I am looking for the name/type of following equations: $$\dot{\theta}\dot{J} = \ddot{x} - J\ddot{\theta}$$ here the unknown is $J \in R^{m \times n}$, $x \in R^{m \times 1}$, $\theta \in R^{n \times 1}$. I've found this equation is same type with 1st order non-homogeneous matrix differential equation: $$dx/dt = Ax + b$$
However, in this case, $x$ unknown is a vector, So my original equation should need some rearrangement
I frequently need to deal with such matrix equations. I am wondering if there is any reference on such matrix rearrangement. It will be more thorough if I can have a general view of these solutions, not just to pick up suggestions from SE at one specific problem
Thanks a lot!
Edit01: one solution
$ \dot{J}={(\ddot{x}-J\ddot{\theta})\dot{\theta}^TW(t)}/\dot{\theta}^TW(t)\dot{\theta} $