System of differential equations

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

Linear system of differential equations
Main article: Linear differential equation

Like any system of equations, a system of linear differential equations is said to be overdetermined if there are more equations than the unknowns. A system of Cauchy–Riemann equations is an example of an overdetermined system.

For an overdetermined system to have a solution, it needs to satisfy the compatibility conditions.[1] For example, consider the system:

\( {\displaystyle {\frac {\partial u}{\partial x_{i}}}=f_{i},1\leq i\leq m.} \)

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

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

See also: Cauchy problem and Ehrenpreis's fundamental principle.
Non-linear system of differential equations
Main article: Nonlinear differential equation

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.)

See also: h-principle.
Differential system

A differential system is a means of studying a system of partial differential equations using geometric ideas such as differential forms and vector fields.

For example, the compatibility conditions of an overdetermined system of differential equations can be succinctly stated in terms of differential forms (i.e., a form to be exact, it needs to be closed). See integrability conditions for differential systems for more.

See also: Category:differential systems.
Notes

https://www.encyclopediaofmath.org/index.php/Overdetermined_system