Variation of Parameters - Description of Method

Description of Method

Given an ordinary non-homogeneous linear differential equation of order n

(i)

let be a fundamental system of solutions of the corresponding homogeneous equation

(ii)

Then a particular solution to the non-homogeneous equation is given by

(iii)

where the are continuous functions which satisfy the equations

(iv)

By substituting (iii) into (i) and applying (iv) it follows that

(v)


The linear system (v) of n equations can then be solved using Cramer's rule yielding

where is the Wronskian determinant of the fundamental system and is the Wronskian determinant of the fundamental system with the i-th column replaced by

The particular solution to the non-homogeneous equation can then be written as

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