In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials.
It is commonly used to solve non-exact ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field).
This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.
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, where
Performing a separation of variables will give This is an implicit solution which involves a nonelementary integral.
This same method is used to solve the period of a simple pendulum.
Integrating factors are useful for solving ordinary differential equations that can be expressed in the form The basic idea is to find some function, say
, called the "integrating factor", which we can multiply through our differential equation in order to bring the left-hand side under a common derivative.
For the canonical first-order linear differential equation shown above, the integrating factor is
Firstly, we only need one integrating factor to solve the equation, not all possible ones; secondly, such constants and absolute values will cancel out even if included.
refers to the sign function, which will be constant on an interval if
, and a logarithm in the antiderivative only appears when the original function involved a logarithm or a reciprocal (neither of which are defined for 0), such an interval will be the interval of validity of our solution.
be the integrating factor of a first order linear differential equation such that multiplication by
, which is a separable differential equation, whose solution yields
gives By applying the product rule in reverse, we see that the left-hand side can be expressed as a single derivative in
We use this fact to simplify our expression to Integrating both sides with respect to
Moving the exponential to the right-hand side, the general solution to Ordinary Differential Equation is: In the case of a homogeneous differential equation,
we obtain The above equation can be rewritten as By integrating both sides with respect to x we obtain or The same result may be achieved using the following approach Reversing the quotient rule gives or or where
The method of integrating factors for first order equations can be naturally extended to second order equations as well.
The main goal in solving first order equations was to find an integrating factor
to work as an integrating factor, then This implies that a second order equation must be exactly in the form
For example, the differential equation can be solved exactly with integrating factors.
This gives us which can be rearranged to give Integrating twice yields Dividing by the integrating factor gives: A slightly less obvious application of second order integrating factors involves the following differential equation: At first glance, this is clearly not in the form needed for second order integrating factors.
However, and from the Pythagorean identity relating cotangent and cosecant, so we actually do have the required term in front of
gives which rearranged is Integrating twice gives Finally, dividing by the integrating factor gives Integrating factors can be extended to any order, though the form of the equation needed to apply them gets more and more specific as order increases, making them less useful for orders 3 and above.
The general idea is to differentiate the function
th order differential equation and combine like terms.
times, dividing by the integrating factor on both sides to achieve the final result.
A third order usage of integrating factors gives thus requiring our equation to be in the form For example in the differential equation we have