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

Read more about this topic:  Variation Of Parameters

Famous quotes containing the words description of, description and/or method:

    The next Augustan age will dawn on the other side of the Atlantic. There will, perhaps, be a Thucydides at Boston, a Xenophon at New York, and, in time, a Virgil at Mexico, and a Newton at Peru. At last, some curious traveller from Lima will visit England and give a description of the ruins of St. Paul’s, like the editions of Balbec and Palmyra.
    Horace Walpole (1717–1797)

    I fancy it must be the quantity of animal food eaten by the English which renders their character insusceptible of civilisation. I suspect it is in their kitchens and not in their churches that their reformation must be worked, and that Missionaries of that description from [France] would avail more than those who should endeavor to tame them by precepts of religion or philosophy.
    Thomas Jefferson (1743–1826)

    Too poor for a bribe, and too proud to importune,
    He had not the method of making a fortune.
    Thomas Gray (1716–1771)