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:
In mathematics, the method of characteristics is a technique for solving partial differential equations.Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation.
This polynomial is called the characteristic polynomial of A. Equation is called the characteristic equation or the secular equation of A. The fundamental theorem of algebra implies that the characteristic polynomial of an n-by-n matrix A, being a polynomial of degree n, can be factored into the product of n linear terms,
The characteristic function of a real-valued random variable always exists, since it is an integral of a bounded continuous function over a space whose measure is finite. A characteristic function is uniformly continuous on the entire space. It is non-vanishing in a region around zero: φ(0) = 1. It is bounded: | φ(t) | ≤ 1.
As a further example, when considering = = (), then the characteristic polynomial is p(x) = x 2 + 1, and the eigenvalues are λ = ±i. As before, evaluating the function at the eigenvalues gives us the linear equations e it = c 0 + i c 1 and e − it = c 0 − ic 1 ; the solution of which gives, c 0 = ( e it + e − it )/2 = cos t and c 1 = ( e ...
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
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 .
This characteristic equation is a nonlinear eigenproblem and there are many methods to compute the spectrum numerically. [7] [8] In some special situations it is possible to solve the characteristic equation explicitly. Consider, for example, the following DDE: = ().