Characteristic equation (calculus)

In mathematics, the characteristic equation (or auxiliary equation^[1]) is an algebraic equation of degree n upon which depends the solution of a given nth-order differential equation^[2] or difference equation.^[3][4] The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients.^[1] Such a differential equation, with y as the dependent variable, superscript (n) denoting n^th-derivative, and a_n, a_n − 1, ..., a₁, a₀ as constants,

${\displaystyle a_{n}y^{(n)}+a_{n-1}y^{(n-1)}+\cdots +a_{1}y'+a_{0}y=0,}$

will have a characteristic equation of the form

${\displaystyle a_{n}r^{n}+a_{n-1}r^{n-1}+\cdots +a_{1}r+a_{0}=0}$

whose solutions r₁, r₂, ..., r_n are the roots from which the general solution can be formed.^[1][5][6] Analogously, a linear difference equation of the form

${\displaystyle y_{t+n}=b_{1}y_{t+n-1}+\cdots +b_{n}y_{t}}$

has characteristic equation

${\displaystyle r^{n}-b_{1}r^{n-1}-\cdots -b_{n}=0,}$

discussed in more detail at Linear recurrence with constant coefficients.

The characteristic roots (roots of the characteristic equation) also provide qualitative information about the behavior of the variable whose evolution is described by the dynamic equation. For a differential equation parameterized on time, the variable's evolution is stable if and only if the real part of each root is negative. For difference equations, there is stability if and only if the modulus of each root is less than 1. For both types of equation, persistent fluctuations occur if there is at least one pair of complex roots.

The method of integrating linear ordinary differential equations with constant coefficients was discovered by Leonhard Euler, who found that the solutions depended on an algebraic 'characteristic' equation.^[2] The qualities of the Euler's characteristic equation were later considered in greater detail by French mathematicians Augustin-Louis Cauchy and Gaspard Monge.^[2][6]