Integrating Factor - General Use

General Use

An integrating factor is any expression that a differential equation is multiplied by to facilitate integration and is not restricted to first order linear equations. For example, the nonlinear second order equation

admits as an integrating factor:

To integrate, note that both sides of the equation may be expressed as derivatives by going backwards with the chain rule:

Therefore

This form may be more useful, depending on application. Performing a separation of variables will give:

this is an implicit solution which involves a nonelementary integral. Though likely too obscure to be useful, this is a general solution. Also, because the previous equation is first order, it could be used for numeric solution in favor of the original equation.

Read more about this topic:  Integrating Factor

Famous quotes containing the word general:

    There is absolutely no evidence—developmental or otherwise—to support separating twins in school as a general policy. . . . The best policy seems to be no policy at all, which means that each year, you and your children need to decide what will work best for you.
    Pamela Patrick Novotny (20th century)

    In the drawing room [of the Queen’s palace] hung a Venus and Cupid by Michaelangelo, in which, instead of a bit of drapery, the painter has placed Cupid’s foot between Venus’s thighs. Queen Caroline asked General Guise, an old connoisseur, if it was not a very fine piece? He replied “Madam, the painter was a fool, for he has placed the foot where the hand should be.”
    Horace Walpole (1717–1797)