Ads
related to: characteristic root equation worksheet 5th
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 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 corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor (possibly negative). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. Its eigenvectors are those ...
This requires some care in the presence of multiple roots; but a complex root and its conjugate do have the same multiplicity (and this lemma is not hard to prove). It can also be worked around by considering only irreducible polynomials ; any real polynomial of odd degree must have an irreducible factor of odd degree, which (having no multiple ...
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.
Even for the first root that involves at most two square roots, the expression of the solutions in terms of radicals is usually highly complicated. However, when no square root is needed, the form of the first solution may be rather simple, as for the equation x 5 − 5x 4 + 30x 3 − 50x 2 + 55x − 21 = 0, for which the only real solution is
An antipalindromic polynomial over a field k with odd characteristic is a multiple of x – 1 (it has 1 as a root) and its quotient by x – 1 is palindromic. An antipalindromic polynomial of even degree is a multiple of x 2 – 1 (it has −1 and 1 as roots) and its quotient by x 2 – 1 is palindromic.
Hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of stable linear systems. Whether a polynomial is Hurwitz can be determined by solving the equation to find the roots, or from the coefficients without solving the equation by the Routh–Hurwitz stability criterion.
Ads
related to: characteristic root equation worksheet 5thteacherspayteachers.com has been visited by 100K+ users in the past month