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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]
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.
The method is based on finite differences where the differentiation operators exhibit summation-by-parts properties. Typically, these operators consist of differentiation matrices with central difference stencils in the interior with carefully chosen one-sided boundary stencils designed to mimic integration-by-parts in the discrete setting.
The designer of the method chooses the coefficients, balancing the need to get a good approximation to the true solution against the desire to get a method that is easy to apply. Often, many coefficients are zero to simplify the method. One can distinguish between explicit and implicit methods.
The coefficients within the major part of the region being integrated are one with non-unit coefficients only at the edges. These two rules can be associated with Euler–MacLaurin formula with the first derivative term and named First order Euler–MacLaurin integration rules . [ 8 ]
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that ...
— Method of undetermined coefficients#Description of the method Is there a more compact way of expressing that second hypothesis? For clarity, the current spelled-out version could be given too as an explanation of a more compact term for such a function, but if there is a more compact term for that mouthful I, as a reader, would appreciate ...
There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only [ 4 ] [ 5 ] (i.e., not its derivatives).