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 systems of differential equations

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. These may be written as

${\frac {dx_{j}}{dt}}=a_{j1}(t)x_{1}+\ldots +a_{jn}(t)x_{n}+g_{j}(t),\qquad j=1,\ldots ,n$

where $n$ is a positive integer, and $a_{ji}(t),g_{j}(t)$ are arbitrary functions of the independent variable t. A first-order linear system of ODEs may be written in matrix form:

${\frac {d}{dt}}{\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}={\begin{bmatrix}a_{11}&\ldots &a_{1n}\\a_{21}&\ldots &a_{2n}\\\vdots &\ldots &\vdots \\a_{n1}&&a_{nn}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}+{\begin{bmatrix}g_{1}\\g_{2}\\\vdots \\g_{n}\end{bmatrix}},$

or simply

$\mathbf {\dot {x}} (t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {g} (t)$ .

Homogeneous systems of differential equations

A linear system is said to be homogeneous if $g_{j}(t)=0$ for each $j$ and for all values of $t$ , otherwise it is referred to as non-homogeneous. Homogeneous systems have the property that if $\mathbf {x_{1}} ,\ldots ,\mathbf {x_{p}}$ are linearly independent solutions to the system, then any linear combination of these, $C_{1}\mathbf {x_{1}} +\ldots +C_{p}\mathbf {x_{p}}$ , is also a solution to the linear system where $C_{1},\ldots ,C_{p}$ are constant.

The case where the coefficients $a_{ji}(t)$ are all constant has a general solution: $\mathbf {x} =C_{1}\mathbf {v_{1}} e^{\lambda _{1}t}+\ldots +C_{n}\mathbf {v_{n}} e^{\lambda _{n}t}$ , where $\lambda _{i}$ is an eigenvalue of the matrix $\mathbf {A}$ with corresponding eigenvectors $\mathbf {v} _{i}$ for $1\leq i\leq n$ . This general solution only applies in cases where $\mathbf {A}$ has n distinct eigenvalues, cases with fewer distinct eigenvalues must be treated differently.

Linear independence of solutions

For an arbitrary system of ODEs, a set of solutions $\mathbf {x_{1}} (t),\ldots ,\mathbf {x_{n}} (t)$ are said to be linearly-independent if:

$C_{1}\mathbf {x_{1}} (t)+\ldots +C_{n}\mathbf {x_{n}} =0\quad \forall t$ is satisfied only for $C_{1}=\ldots =C_{n}=0$ .

A second-order differential equation ${\ddot {x}}=f(t,x,{\dot {x}})$ may be converted into a system of first order linear differential equations by defining $y={\dot {x}}$ , which gives us the first-order system:

${\begin{cases}{\dot {x}}&=&y\\{\dot {y}}&=&f(t,x,y)\end{cases}}$

Just as with any linear system of two equations, two solutions may be called linearly-independent if $C_{1}\mathbf {x} _{1}+C_{2}\mathbf {x} _{2}=\mathbf {0}$ implies $C_{1}=C_{2}=0$ , or equivalently that ${\begin{vmatrix}x_{1}&x_{2}\\{\dot {x}}_{1}&{\dot {x}}_{2}\end{vmatrix}}$ is non-zero. 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.

Overdetermination of systems 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.^[1] 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.)

Other examples of non-linear systems of differential equations include the Lotka–Volterra equations.

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.

Notes

^ "Overdetermined system - Encyclopedia of Mathematics".

References

L. Ehrenpreis, The Universality of the Radon Transform, Oxford Univ. Press, 2003.
Gromov, M. (1986), Partial differential relations, Springer, ISBN 3-540-12177-3
M. Kuranishi, "Lectures on involutive systems of partial differential equations", Publ. Soc. Mat. São Paulo (1967)
Pierre Schapira, Microdifferential systems in the complex domain, Grundlehren der Math- ematischen Wissenschaften, vol. 269, Springer-Verlag, 1985.