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g(x) is a constant, a polynomial function, exponential function , sine or cosine functions or , or finite sums and products of these functions (, constants). The method consists of finding the general homogeneous solution y c {\displaystyle y_{c}} for the complementary linear homogeneous differential equation
[3] [4] The characteristic equation can only be formed when the differential equation is linear and homogeneous, and has constant coefficients. [1] Such a differential equation, with y as the dependent variable, superscript (n) denoting n th-derivative, and a n, a n − 1, ..., a 1, a 0 as constants,
Thus solving a polynomial system over a number field is reduced to solving another system over the rational numbers. For example, if a system contains 2 {\displaystyle {\sqrt {2}}} , a system over the rational numbers is obtained by adding the equation r 2 2 – 2 = 0 and replacing 2 {\displaystyle {\sqrt {2}}} by r 2 in the other equations.
In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. The set of the solutions of such a system is a differential algebraic variety , and corresponds to an ideal in a differential algebra of differential ...
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...
The analytical method of separation of variables for solving partial differential equations has also been generalized into a computational method of decomposition in invariant structures that can be used to solve systems of partial differential equations. [1]
The Characteristic Set Method is the first factorization-free algorithm, which was proposed for decomposing an algebraic variety into equidimensional components. Moreover, the Author, Wen-Tsun Wu, realized an implementation of this method and reported experimental data in his 1987 pioneer article titled "A zero structure theorem for polynomial equations solving". [1]
Frequently models of physical systems contain terms representing fast-decaying elements (i.e. with large negative exponential arguments). Even when these are not of interest in the overall solution, the instability they can induce means that an exceptionally small timestep would be required if the Euler method is used.