Search results
Results From The WOW.Com Content Network
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 θ + i sin θ, this solution can be rewritten as follows:
The first order autoregressive model, = +, has a unit root when =.In this example, the characteristic equation is =.The root of the equation is =.. If the process has a unit root, then it is a non-stationary time series.
As an example, consider the advection equation (this example assumes familiarity with PDE notation, and solutions to basic ODEs). + = where is constant and is a function of and . We want to transform this linear first-order PDE into an ODE along the appropriate curve; i.e. something of the form
Characteristic equation may refer to: Characteristic equation (calculus), used to solve linear differential equations; Characteristic equation, the equation obtained by equating to zero the characteristic polynomial of a matrix or of a linear mapping; Method of characteristics, a technique for solving partial differential equations
In mathematics, and particularly ordinary differential equations, a characteristic multiplier is an eigenvalue of a monodromy matrix. The logarithm of a characteristic multiplier is also known as characteristic exponent. [1] They appear in Floquet theory of periodic differential operators and in the Frobenius method.
The characteristic equation, also known as the determinantal equation, [1] [2] [3] is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory , the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix .
The importance of the criterion is that the roots p of the characteristic equation of a linear system with negative real parts represent solutions e pt of the system that are stable . Thus the criterion provides a way to determine if the equations of motion of a linear system have only stable solutions, without solving the system directly.
In mathematics, the term "characteristic function" can refer to any of several distinct concepts: The indicator function of a subset , that is the function 1 A : X → { 0 , 1 } , {\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\},} which for a given subset A of X , has value 1 at points of A and 0 at points of X − A .