Characteristic Equation (calculus)
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In mathematics, the characteristic equation (or auxiliary equation) is an algebraic equation of
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upon which depends the solution of a given th-
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
or
difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramete ...
. The characteristic equation can only be formed when the differential or difference equation is
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
, and has constant
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s. Such a differential equation, with as the
dependent variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or dema ...
, superscript denoting ''n''th-derivative, and as
constants Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable or scientific const ...
, :a_y^ + a_y^ + \cdots + a_y' + a_y = 0, will have a characteristic equation of the form :a_r^ + a_r^ + \cdots + a_r + a_ = 0 whose solutions are the roots from which the
general solution In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...
can be formed. Analogously, a linear difference equation of the form :y_=b_1y_ + \cdots + b_ny_ has characteristic equation :r^n - b_1r^ - \cdots - b_n =0, 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 Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, who found that the solutions depended on an algebraic 'characteristic' equation. The qualities of the Euler's characteristic equation were later considered in greater detail by French mathematicians
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...
and
Gaspard Monge Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. Duri ...
.


Derivation

Starting with a linear homogeneous differential equation with constant coefficients , :a_n y^ + a_y^ + \cdots + a_1 y^\prime + a_0 y = 0 it can be seen that if , each term would be a constant multiple of . This results from the fact that the derivative of the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
is a multiple of itself. Therefore, , , and are all multiples. This suggests that certain values of will allow multiples of to sum to zero, thus solving the homogeneous differential equation. In order to solve for , one can substitute and its derivatives into the differential equation to get :a_n r^n e^ + a_r^e^ + \cdots + a_1 re^ + a_0 e^ = 0 Since can never equal zero, it can be divided out, giving the characteristic equation :a_n r^n + a_r^ + \cdots + a_1 r + a_0 = 0 By solving for the roots, , in this characteristic equation, one can find the general solution to the differential equation. For example, if has roots equal to , then the general solution will be y(x) = c_1 e^ + c_2 e^ + c_3 e^, where c_1, c_2, and c_3 are arbitrary constants which need to be determined by the boundary and/or initial conditions.


Formation of the general solution

Solving the characteristic equation for its roots, , 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, repeated roots, or complex roots corresponding to general solutions of , , and , respectively, then the general solution to the differential equation is : y(x) = y_\mathrm(x) + y_(x) + \cdots + y_(x) + y_(x) + \cdots + y_(x)


Example

The linear homogeneous differential equation with constant coefficients : y^ + y^ - 4y^ - 16y'' -20y' - 12y = 0 has the characteristic equation : r^5 + r^4 - 4r^3 - 16r^2 -20r - 12 = 0 By factoring the characteristic equation into : (r - 3)\left(r^2 + 2r + 2\right)^2 = 0 one can see that the solutions for are the distinct single root and the double complex roots . This corresponds to the real-valued general solution : y(x) = c_1 e^ + e^x(c_2 \cos x + c_3 \sin x) + xe^x(c_4 \cos x + c_5 \sin x) with constants .


Distinct real roots

The ''
superposition principle The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So th ...
for linear homogeneous differential equations with constant coefficients'' says that if are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ...
solutions to a particular differential equation, then is also a solution for all values . Therefore, if the characteristic equation has distinct real roots , then a general solution will be of the form : y_\mathrm(x) = c_1 e^ + c_2 e^ + \cdots + c_n e^


Repeated real roots

If the characteristic equation has a root that is repeated times, then it is clear that is at least one solution. However, this solution lacks linearly independent solutions from the other roots. Since has multiplicity , the differential equation can be factored into : \left ( \frac - r_1 \right )^k y = 0 . The fact that is one solution allows one to presume that the general solution may be of the form , where is a function to be determined. Substituting gives : \left ( \frac - r_1 \right ) ue^ = \frac\left(ue^\right) - r_1 ue^ = \frac(u)e^ + r_1 ue^- r_1 ue^ = \frac(u)e^ when . By applying this fact times, it follows that : \left ( \frac - r_1 \right )^k ue^ = \frac(u)e^ = 0 By dividing out , it can be seen that : \frac(u) = u^ = 0 Therefore, the general case for is a polynomial of degree , so that . Since , the part of the general solution corresponding to is : y_\mathrm(x) = e^\left(c_1 + c_2 x + \cdots + c_k x^\right)


Complex roots

If a second-order differential equation has a characteristic equation with complex conjugate roots of the form and , then the general solution is accordingly . By
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for ...
, which states that , this solution can be rewritten as follows: :\begin y(x) &= c_e^ + c_e^\\ &= c_e^(\cos bx + i \sin bx) + c_e^( \cos bx - i \sin bx ) \\ &= \left(c_ + c_\right)e^ \cos bx + i(c_ - c_)e^ \sin bx \end where and are constants that can be non-real and which depend on the initial conditions. (Indeed, since is real, must be imaginary or zero and must be real, in order for both terms after the last equality sign to be real.) For example, if , then the particular solution is formed. Similarly, if and , then the independent solution formed is . Thus by the ''superposition principle for linear homogeneous differential equations with constant coefficients'', a second-order differential equation having complex roots will result in the following general solution: : y_\mathrm(x) = e^\left(C_ \cos bx +C_ \sin bx \right) 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.


See also

*
Characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...


References

{{reflist, 2 Ordinary differential equations