Search results
Results from the WOW.Com Content Network
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]
These results were later generalized to spatially homogeneous random media modeled by differential equations with random coefficients which statistical properties are the same at every point in space. [5] [6] In practice, many applications require a more general way of modeling that is neither periodic nor statistically homogeneous. For this ...
If a system initially rests at its equilibrium position, from where it is acted upon by a unit-impulse at the instance t=0, i.e., p(t) in the equation above is a Dirac delta function δ(t), () = | = =, then by solving the differential equation one can get a fundamental solution (known as a unit-impulse response function)
Definition Field of application Coefficient of determination: statistics (proportion of variance explained by a statistical model) Coefficient of variation: statistics (ratio of standard deviation to expectation) Correlation: ρ or r
Consider the general, homogeneous, second-order linear constant coefficient ordinary differential equation. (ODE) ″ + ′ + =, where ,, are real non-zero coefficients. . Two linearly independent solutions for this ODE can be straightforwardly found using characteristic equations except for the case when the discriminant, , vanish
A coefficient is a constant coefficient when it is a constant function. For avoiding confusion, in this context a coefficient that is not attached to unknown functions or their derivatives is generally called a constant term rather than a constant coefficient. In particular, in a linear differential equation with constant coefficient, the ...
In a non-homogeneous medium, these parameters can vary with altitude and location along the path, formally making these terms n(s), σ λ (s), T(s), and I λ (s). Additional terms are added when scattering is important. Integrating the change in spectral intensity [W/sr/m 2 /μm] over all relevant wavelengths gives the change in intensity [W/sr ...
In statistics, homogeneity and its opposite, heterogeneity, arise in describing the properties of a dataset, or several datasets.They relate to the validity of the often convenient assumption that the statistical properties of any one part of an overall dataset are the same as any other part.