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Consider the following methods of finding the particular integral for a non-homogeneous differential equation. Discuss the reasoning and answers of the given questions.
(a) Methods of undetermined coefficient (Superposition approach and Annihilator approach)
(b) Method of variation of parameters.
1) For which particular forms of the input function g(x), the methods of undetermined coefficients are used? Also, discuss the reason to restrict to those particular forms of g(x).
2) In what way do you think the method of variation of parameters has an advantage over the methods of undetermined coefficients.
what way do you think the method of variation of parameters has an advantage over the methods of undetermined coefficients discuss PLZ
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The input function can have one of the following forms: )(xg
A constant function k.
A polynomial function
An exponential function e
The trigonometric functions ) cos( ), sin( xxβ β
Finite sums and products of these functions.
Otherwise, we cannot apply the method of undetermined coefficients
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The input function gis restricted to have one of the above stated forms because of the
The derivatives of sums and products of polynomials, exponentials etc are again
sums and products of similar kind of functions.
The expression has to be identically equal to the input
ay ++/ //
Therefore, to make an educated guess, is assured to have the same form as