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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]
When the non-homogeneous term is expressed as an exponential function, the ERF method or the undetermined coefficients method can be used to find a particular solution. If non-homogeneous terms can not be transformed to complex exponential function, then the Lagrange method of variation of parameters can be used to find solutions.
In mathematics, the annihilator method is a procedure used to find a particular solution to certain types of non-homogeneous ordinary differential equations (ODEs). [1] It is similar to the method of undetermined coefficients, but instead of guessing the particular solution in the method of undetermined coefficients, the particular solution is determined systematically in this technique.
One partial solution of the non-homogeneous equation: () + ¯ + ¯ = ¯ (), where ¯ = (), could be obtained by the Lagrangian method for deriving partial solution of non-homogeneous ordinary differential equations.
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 ...
Intuitively, one can think of the inhomogeneous problem as a set of homogeneous problems each starting afresh at a different time slice t = t 0. By linearity, one can add up (integrate) the resulting solutions through time t 0 and obtain the solution for the inhomogeneous problem. This is the essence of Duhamel's principle.
The formula for the exponential results from reducing the powers of G in the series expansion and identifying the respective series coefficients of G 2 and G with −cos(θ) and sin(θ) respectively. The second expression here for e Gθ is the same as the expression for R ( θ ) in the article containing the derivation of the generator , R ( θ ...
Of course, all matter is inhomogeneous at some scale, but frequently it is convenient to treat it as homogeneous. A good example is the continuum concept which is used in continuum mechanics . Under this assumption, materials such as fluids , solids , etc. can be treated as homogeneous materials and associated with these materials are material ...