# System of differential equations

In mathematics, a system of differential equations is a finite set of differential equations. Such a system can be either linear or non-linear. Also, such a system can be either a system of ordinary differential equations or a system of partial differential equations.

## Linear system of differential equations

Like any system of equations, a system of linear differential equations is said to be overdetermined if there are more equations than the unknowns.

For an overdetermined system to have a solution, it needs to satisfy the compatibility conditions. For example, consider the system:

${\frac {\partial u}{\partial x_{i}}}=f_{i},1\leq i\leq m.$

Then the necessary conditions for the system to have a solution are:

${\frac {\partial f_{i}}{\partial x_{k}}}-{\frac {\partial f_{k}}{\partial x_{i}}}=0,1\leq i,k\leq m.$

## Non-linear system of differential equations

Perhaps the most famous example of a non-linear system of differential equations is the Navier–Stokes equations. Unlike the linear case, the existence of a solution of a non-linear system is a difficult problem (cf. Navier–Stokes existence and smoothness.)