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:
“We have seen over and over that white male historians in general have tended to dismiss any history they didn’t themselves write, on the grounds that it is unserious, unscholarly, a fad, too “political,” “merely” oral and thus unreliable.”
—Adrienne Rich (b. 1929)
“The most general deficiency in our sort of culture and education is gradually dawning on me: no one learns, no one strives towards, no one teaches—enduring loneliness.”
—Friedrich Nietzsche (1844–1900)