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:
“I was here first introduced to Joe.... He was a good-looking Indian, twenty-four years old, apparently of unmixed blood, short and stout, with a broad face and reddish complexion, and eyes, methinks, narrower and more turned up at the outer corners than ours, answering to the description of his race. Besides his underclothing, he wore a red flannel shirt, woolen pants, and a black Kossuth hat, the ordinary dress of the lumberman, and, to a considerable extent, of the Penobscot Indian.”
—Henry David Thoreau (1817–1862)
“Whose are the truly labored sentences? From the weak and flimsy periods of the politician and literary man, we are glad to turn even to the description of work, the simple record of the month’s labor in the farmer’s almanac, to restore our tone and spirits.”
—Henry David Thoreau (1817–1862)
“Government by average opinion is merely a circuitous method of going to the devil; those who profess to lead but in fact slavishly follow this average opinion are simply the fastest runners and the loudest squeakers of the herd which is rushing blindly down to its destruction.”
—Thomas Henry Huxley (1825–95)