Method that combine LinearSolve and Dsolve for a set of differential equations?

Question

I have a matrix for example: $\begin{bmatrix} 0 & 5 & 0 & 3 & 3\\ 5 & 5 & 5 & 3 & 1\\ 0 & 3 & 0 & 3 & 10\\ 1 & 3 & 2 & 3 & 10 \\ 1 & 2 & 3 & 4 & 5 \end{bmatrix}$ representing a set of differential equations: $\begin{cases} 5x_{2} + 3x_{4} + 3x_{5} = \frac{dx_{1}}{dt}\\ 5x_{1} + 5x_{2} + 5x_{3} + 3x_{4} + x_{5}= 0\\ 3x_{2} + 3x_{4} + 10x_{5} = \frac{dx_{3}}{dt}\\x_{1} + 3_{2} + 2x_{3} + 3x_{4} + 10x_{5} = 0\\ x_{1} + 2x_{2} + 3x_{3} + 4x_{4} + 5x_{5} = \frac{dx_{5}}{dt}\end{cases}$ where $\frac{dx_{2}}{dt}=0$ and $\frac{dx_{4}}{dt}=0$

Dsolve cannot take the case of $\frac{dx_{2}}{dt}=0$ and $\frac{dx_{4}}{dt}=0$; while LinearSolve cannot solve differential equations. How can I incorporate the two methods to solve a set of differential equations like that? It would be great if the input of matrix (which can be used in DSolve and LinearSolve) can be used conveniently to represent such system (without typing the whole set of equations as I have about 50 equations). For example, http://reference.wolfram.com/language/tutorial/DSolveSystemsOfLinearODEs.html

Can someone help me? Thank you very much.

(Note: The above set of differential equations is just arbitrary, may not be consistent or solvable.)

Answer 1

Here's my interpretation:

SeedRandom[0];
matrix = RandomInteger[10, {5, 5}];
zero = {2, 4};  (* equations to set to zero *)
eqns = matrix.Array[x[#][t] &, 5] == Array[x[#]'[t] &, {5}] /. 
   x[Alternatives @@ zero]'[t] -> 0;
eqns /. x[i_Integer] :> Subscript[x, i] // Thread // MatrixForm

{sol0} = DSolve[eqns, Array[x, 5], t];
sol0 // Short