Linear differential equation

In mathematics, linear differential equations are differential equations having solutions which can be added together in particular linear combinations to form further solutions. They can be ordinary (ODEs) or partial (PDEs). The solutions to (homogeneous) linear differential equations form a vector space (unlike non-linear differential equations).

Introduction

Linear differential equations are of the form

where the differential operator L is a linear operator, y is the unknown function (such as a function of time y(t)), and the right hand side f is a given function of the same nature as y (called the source term). For a function dependent on time we may write the equation more expressly as

and, even more precisely by bracketing

The linear operator L may be considered to be of the form

The linearity condition on L rules out operations such as taking the square of the derivative of y; but permits, for example, taking the second derivative of y. It is convenient to rewrite this equation in an operator form

Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.

If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.