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.