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Consider a linear non-homogeneous ordinary differential equation of the form = + (+) = where () denotes the i-th derivative of , and denotes a function of .. The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met: [2]
A variable may denote an unknown number that has to be determined; in which case, it is called an unknown; for example, in the quadratic equation ax 2 + bx + c = 0, the variables a, b, c are parameters, and x is the unknown. Sometimes the same symbol can be used to denote both a variable and a constant, that is a well defined mathematical object.
WolframAlpha (/ ˈ w ʊ l f. r əm-/ WUULf-rəm-) is an answer engine developed by Wolfram Research. [3] It is offered as an online service that answers factual queries by computing answers from externally sourced data. [4] [5]
Applicable to: square, hermitian, positive definite matrix Decomposition: =, where is upper triangular with real positive diagonal entries Comment: if the matrix is Hermitian and positive semi-definite, then it has a decomposition of the form = if the diagonal entries of are allowed to be zero
All primes after 2 are odd. Even numbers are always 0, 2, or 4 more than a multiple of 6, while odd numbers are always 1, 3, or 5 more than a multiple of 6. ... Wolfram Alpha. ... But somehow it ...
[4] [5] [6] Cramer's rule, implemented in a naive way, is computationally inefficient for systems of more than two or three equations. [7] In the case of n equations in n unknowns, it requires computation of n + 1 determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single ...
This equation is a special form of the more general weakly singular Volterra integral equation of the first kind, called Abel's integral equation: [7] = Strongly singular: An integral equation is called strongly singular if the integral is defined by a special regularisation, for example, by the Cauchy principal value.
The notation convention chosen here (with W 0 and W −1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. [3]The name "product logarithm" can be understood as follows: since the inverse function of f(w) = e w is termed the logarithm, it makes sense to call the inverse "function" of the product we w the "product logarithm".