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.
[1] A first-order linear system of ODEs is a system in which every equation is first order and depends on the unknown functions linearly.
Here we consider systems with an equal number of unknown functions and equations.
{\displaystyle {\frac {dx_{j}}{dt}}=a_{j1}(t)x_{1}+\ldots +a_{jn}(t)x_{n}+g_{j}(t),\qquad j=1,\ldots ,n}
is a positive integer, and
are arbitrary functions of the independent variable t. A first-order linear system of ODEs may be written in matrix form:
A linear system is said to be homogeneous if
Homogeneous systems have the property that if
are linearly independent solutions to the system, then any linear combination of these,
, is also a solution to the linear system where
are all constant has a general solution:
This general solution only applies in cases where
has n distinct eigenvalues, cases with fewer distinct eigenvalues must be treated differently.
For an arbitrary system of ODEs, a set of solutions
A second-order differential equation
may be converted into a system of first order linear differential equations by defining
Just as with any linear system of two equations, two solutions may be called linearly-independent if
This notion is extended to second-order systems, and any two solutions to a second-order ODE are called linearly-independent if they are linearly-independent in this sense.
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.
[2] For example, consider the system: Then the necessary conditions for the system to have a solution are: See also: Cauchy problem and Ehrenpreis's fundamental principle.
Perhaps the most famous example of a nonlinear system of differential equations is the Navier–Stokes equations.
Unlike the linear case, the existence of a solution of a nonlinear system is a difficult problem (cf.
Other examples of nonlinear systems of differential equations include the Lotka–Volterra equations.
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., for a form to be exact, it needs to be closed).
See integrability conditions for differential systems for more.
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