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The ERF method of finding a particular solution of a non-homogeneous differential equation is applicable if the non-homogeneous equation is or could be transformed to form () = + + +; where , are real or complex numbers and () is homogeneous linear differential equation of any order. Then, the exponential response formula can be applied to each ...
with the derivative evaluated at = Another connexion with the confluent hypergeometric functions is that E 1 is an exponential times the function U(1,1,z): = (,,) The exponential integral is closely related to the logarithmic integral function li(x) by the formula
Differential quadrature is the approximation of derivatives by using weighted sums of function values. [ 22 ] [ 23 ] Differential quadrature is of practical interest because its allows one to compute derivatives from noisy data.
Class of differential equation which may sometimes be solved exactly [3] Briot-Bouquet Equation: 1 ′ = (,) Class of differential equation which may sometimes be solved exactly [4] Cherwell-Wright differential equation: 1
Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. [ citation needed ] Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction—each of which may lead to a simplified ...
where is the amplitude of Gaussian, = is exponent relaxation time, is a variance of exponential probability density function. This function cannot be calculated for some values of parameters (for example, =) because of arithmetic overflow.
In multivariate calculus, a differential or differential form is said to be exact or perfect (exact differential), as contrasted with an inexact differential, if it is equal to the general differential for some differentiable function in an orthogonal coordinate system (hence is a multivariable function whose variables are independent, as they are always expected to be when treated in ...
The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential dy as an infinitely small (or infinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x.