# real analysis – A system of matrix differential equations : reference request

Consider the system of matrix differential equations.

$$Dfrac{partial{C(t)}}{partial{t}}S_i + frac{1}{n}C(t)Q_iDC(t)S_i – C(t)Q_iE = 0$$ or
$$Ddot{C}(t)S_i + frac{1}{n}C(t)Q_iDC(t)S_i – C(t)Q_iE = 0$$

system consists of total $$n$$ matrix differential equations for $$i=1,2,ldots n$$.

There are $$(m+1)n$$ unknowns and as each equation is a system of $$(m+1)$$ equations and and as there are $$n$$ such systems, total number of equations are $$(m+1)n$$. Hence there are $$(m+1)n$$ unknowns and $$(m+1)n$$ equations.

given initial condition $$C(0) = C_0$$.

Matrix dimensions

$$C(t)$$———-> $$(m+1)times n$$

$$S_i$$————–>$$ntimes 1$$, $$i=1,2,ldots n$$

$$Q_i$$————–>$$ntimes(m+1)$$, $$i=1,2,ldots n$$

$$D$$————–>$$(m+1)times(m+1)$$ diagonal matrix.

$$E$$————–>$$(m+1)times 1$$

If its useful to know, $$n>>m$$ and $$mge 3$$.

I stumbled upon this when I am trying to work on Navier-Stokes Equations. I don’t have any background in the system of matrix equations, and I’d like to know if there is any way to solve this equation.

This is basically related to the simple Bernoulli equation with constant coefficients as $$y’ = ay^2+by$$ which can be solved by substituting $$y = frac{1}{u}$$ and we get the final solution as $$y = -frac{b}{a+e^{-bx}}$$

But here, we have a system of matrix equations (a more general case than an ODE system). I’d like to know any reference for a system of matrix equations and then hopefully a reference for a system of matrix differential equations.
One reference I could find is this, but it seems a linear system. I am hoping that much work is available for the case in this question.