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 are two great rules in life, the one general and the other particular. The first is that every one can in the end get what he wants if he only tries. This is the general rule. The particular rule is that every individual is more or less of an exception to the general rule.”
—Samuel Butler (1835–1902)
“The bond between a man and his profession is similar to that which ties him to his country; it is just as complex, often ambivalent, and in general it is understood completely only when it is broken: by exile or emigration in the case of one’s country, by retirement in the case of a trade or profession.”
—Primo Levi (1919–1987)