Characteristic equation (calculus)
In mathematics, the characteristic equation (or auxiliary equation) is an algebraic equation of degree n upon which depends the solution of a given nth-order differential equation or difference equation. The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients. Such a differential equation, with y as the dependent variable, superscript (n) denoting nth-derivative, and an, an − 1, ..., a1, a0 as constants,
will have a characteristic equation of the form
has characteristic equation
discussed in more detail at Linear difference equation#Solution of 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. The qualities of the Euler's characteristic equation were later considered in greater detail by French mathematicians Augustin-Louis Cauchy and Gaspard Monge.
Starting with a linear homogeneous differential equation with constant coefficients an, an − 1, ..., a1, a0,
it can be seen that if y(x) = erx, each term would be a constant multiple of erx. This results from the fact that the derivative of the exponential function erx is a multiple of itself. Therefore, y′ = rerx, y″ = r2erx, and y(n) = rnerx are all multiples. This suggests that certain values of r will allow multiples of erx to sum to zero, thus solving the homogeneous differential equation. In order to solve for r, one can substitute y = erx and its derivatives into the differential equation to get
Since erx 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. For example, if r is found to equal to 3, then the general solution will be y(x) = ce3x, where c is an arbitrary constant.
Formation of the general solution
Solving the characteristic equation for its roots, r1, ..., rn, 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 yD(x), yR1(x), ..., yRh(x), and yC1(x), ..., yCk(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 r1 = 3 and the double complex roots r2,3,4,5 = −1 ± i. This corresponds to the real-valued general solution
with constants c1, ..., c5.
Distinct real roots
The superposition principle for linear homogeneous differential equations with constant coefficients says that if u1, ..., un are n linearly independent solutions to a particular differential equation, then c1u1 + ... + cnun is also a solution for all values c1, ..., cn. Therefore, if the characteristic equation has distinct real roots r1, ..., rn, then a general solution will be of the form
Repeated real roots
If the characteristic equation has a root r1 that is repeated k times, then it is clear that yp(x) = c1er1x is at least one solution. However, this solution lacks linearly independent solutions from the other k − 1 roots. Since r1 has multiplicity k, the differential equation can be factored into
The fact that yp(x) = c1er1x is one solution allows one to presume that the general solution may be of the form y(x) = u(x)er1x, where u(x) is a function to be determined. Substituting uer1x gives
when k = 1. By applying this fact k times, it follows that
By dividing out er1x, it can be seen that
However, this is the case if and only if u(x) is a polynomial of degree k − 1, so that u(x) = c1 + c2x + c3x2 + ... + ckxk − 1. Since y(x) = uer1x, the part of the general solution corresponding to r1 is
If a second-order differential equation has a characteristic equation with complex conjugate roots of the form r1 = a + bi and r2 = a − bi, then the general solution is accordingly y(x) = c1e(a + bi)x + c2e(a − bi)x. By Euler's formula, which states that eiθ = cos θ + i sin θ, this solution can be rewritten as follows:
where c1 and c2 are constants that can be non-real and which depend on the initial conditions. (Indeed, since y(x) is real, c1 − c2 must be imaginary or zero and c1 + c2 must be real, in order for both terms after the last equality sign to be real.)
For example, if c1 = c2 = 1/, then the particular solution y1(x) = eax cos bx is formed. Similarly, if c1 = 1/ and c2 = −1/, then the independent solution formed is y2(x) = eax sin bx. Thus by the superposition principle for linear homogeneous differential equations with constant coefficients, a second-order differential equation having complex roots r = a ± bi will result in the following general solution:
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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