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 grown literally afraid to be poor. We despise anyone who elects to be poor in order to simplify and save his inner life. If he does not join the general scramble and pant with the money-making street, we deem him spiritless and lacking in ambition.”
—William James (1842–1910)
“We do not need to minimize the poverty of the ghetto or the suffering inflicted by whites on blacks in order to see that the increasingly dangerous and unpredictable conditions of middle- class life have given rise to similar strategies for survival. Indeed the attraction of black culture for disaffected whites suggests that black culture now speaks to a general condition.”
—Christopher Lasch (b. 1932)