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:
“Why not draft executive and management brains to prepare and produce the equipment the $21-a-month draftee must use and forget this dollar-a-year tommyrot? Would we send an army into the field under a dollar-a-year General who had to be home Mondays, Wednesdays and Fridays?”
—Lyndon Baines Johnson (1908–1973)
“As a general rule, do not kick the shins of the opposite gentleman under the table, if personally unaquainted with him; your pleasantry is liable to be misunderstood—a circumstance at all times unpleasant.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)