Matched filter is very common and termed as optimal filter for detection purpose. The question is : can we formulate a differential equation and show that the solution is matched filter.
As the name implies, a matched filter is found to be an exact replica of the signal of interest to be detected; just a reversed version as a result of an optimization.
Hence your question reduces to whether the signal of interest is produced as a solution of some diferential equation.
Even though it's true that there are signals as solutions to differential equations of all sorts (linear, nonlinear, time varying, time invariant etc), whether an arbitrary signal can be imposed to be a solution to some solvable differential equation is not very clear to me.
Since the signal of interest is an arbitrary signal, you must show that every (arbitrary) signal can be shown to be a solution of a corresponing differential equation. I would say no.
Note that as a degenerate case $y(t) = x(t)$ can also be considered to be a (useless) differential equation (of order zero?) but I think it's not what you mean by a differential equation.