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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}, ..., *a*_{1}, *a*_{0} as constants,

will have a characteristic equation of the form

whose solutions *r*_{1}, *r*_{2}, ..., *r*_{n} are the roots from which the general solution can be formed.^{[1]}^{[5]}^{[6]} Analogously, a linear difference equation of the form

has characteristic equation

discussed in more detail at Linear recurrence with constant coefficients#Solution to homogeneous case.

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 (absolute value) 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]}

Starting with a linear homogeneous differential equation with constant coefficients *a*_{n}, *a*_{n − 1}, ..., *a*_{1}, *a*_{0},

it can be seen that if *y*(*x*) = *e ^{rx}*, each term would be a constant multiple of

Since *e ^{rx}* can never equal zero, it can be divided out, giving the characteristic equation

By solving for the roots, r, in this characteristic equation, one can find the general solution to the differential equation.^{[1]}^{[6]} For example, if r has roots equal to {3, 11, 40}, then the general solution will be , where , , and are arbitrary constants which need to be determined by the boundary and/or initial conditions.

Solving the characteristic equation for its roots, *r*_{1}, ..., *r*_{n}, allows one to find the general solution of the differential equation. The roots may be real or complex, as well as distinct or repeated. If a characteristic equation has parts with distinct real roots, h repeated roots, or k complex roots corresponding to general solutions of *y*_{D}(*x*), *y*_{R1}(*x*), ..., *y*_{Rh}(*x*), and *y*_{C1}(*x*), ..., *y*_{Ck}(*x*), respectively, then the general solution to the differential equation is

The linear homogeneous differential equation with constant coefficients

has the characteristic equation

By factoring the characteristic equation into

one can see that the solutions for r are the distinct single root *r*_{1} = 3 and the double complex roots *r*_{2,3,4,5} = 1 ± *i*. This corresponds to the real-valued general solution

with constants *c*_{1}, ..., *c*_{5}.

The *superposition principle for linear homogeneous differential equations with constant coefficients* says that if *u*_{1}, ..., *u*_{n} are n linearly independent solutions to a particular differential equation, then *c*_{1}*u*_{1} + ⋯ + *c _{n}u_{n}* is also a solution for all values

If the characteristic equation has a root *r*_{1} that is repeated k times, then it is clear that *y*_{p}(*x*) = *c*_{1}*e*^{r1x} is at least one solution.^{[1]} However, this solution lacks linearly independent solutions from the other *k* − 1 roots. Since *r*_{1} has multiplicity k, the differential equation can be factored into^{[1]}

The fact that *y*_{p}(*x*) = *c*_{1}*e*^{r1x} is one solution allows one to presume that the general solution may be of the form *y*(*x*) = *u*(*x*)*e*^{r1x}, where *u*(*x*) is a function to be determined. Substituting *ue*^{r1x} gives

when *k* = 1. By applying this fact k times, it follows that

By dividing out *e*^{r1x}, it can be seen that

Therefore, the general case for *u*(*x*) is a polynomial of degree *k*−1, so that *u*(*x*) = *c*_{1} + *c*_{2}*x* + *c*_{3}*x*^{2} + ⋯ + *c _{k}x*

If a second-order differential equation has a characteristic equation with complex conjugate roots of the form *r*_{1} = *a* + *bi* and *r*_{2} = *a* − *bi*, then the general solution is accordingly *y*(*x*) = *c*_{1}*e*^{(a + bi)x} + *c*_{2}*e*^{(a − bi)x}. By Euler's formula, which states that *e ^{iθ}* = cos

where *c*_{1} and *c*_{2} are constants that can be non-real and which depend on the initial conditions.^{[6]} (Indeed, since *y*(*x*) is real, *c*_{1} − *c*_{2} must be imaginary or zero and *c*_{1} + *c*_{2} must be real, in order for both terms after the last equality sign to be real.)

For example, if *c*_{1} = *c*_{2} = 1/2, then the particular solution *y*_{1}(*x*) = *e ^{ax}* cos

This analysis also applies to the parts of the solutions of a higher-order differential equation whose characteristic equation involves non-real complex conjugate roots.

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^{a}^{b}^{c}^{d}^{e}^{f}^{g}Edwards, C. Henry; Penney, David E. "Chapter 3".*Differential Equations: Computing and Modeling*. David Calvis. Upper Saddle River, New Jersey: Pearson Education. pp. 156–170. ISBN 978-0-13-600438-7. - ^
^{a}^{b}^{c}Smith, David Eugene. "History of Modern Mathematics: Differential Equations". University of South Florida. **^**Baumol, William J. (1970).*Economic Dynamics*(3rd ed.). p. 172.**^**Chiang, Alpha (1984).*Fundamental Methods of Mathematical Economics*(3rd ed.). pp. 578, 600.- ^
^{a}^{b}Chu, Herman; Shah, Gaurav; Macall, Tom. "Linear Homogeneous Ordinary Differential Equations with Constant Coefficients". eFunda. Retrieved 1 March 2011. - ^
^{a}^{b}^{c}^{d}^{e}Cohen, Abraham (1906).*An Elementary Treatise on Differential Equations*. D. C. Heath and Company. **^**Dawkins, Paul. "Differential Equation Terminology".*Paul's Online Math Notes*. Retrieved 2 March 2011.