An integrating factor is any expression that a differential equation is multiplied by to facilitate integration. For example, the nonlinear second order equation
To integrate, note that both sides of the equation may be expressed as derivatives by going backwards with the chain rule:
Solving first order linear ordinary differential equations
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 .
Note that it is not necessary to include the arbitrary constant in the integral, or absolute values in case the integral of involves a logarithm. 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. For absolute values, this can be seen by writing , where refers to the sign function, which will be constant on an interval if is continuous. As is undefined when , 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.
To derive this, let be the integrating factor of a first order linear differential equation such that multiplication by transforms a partial derivative into a total derivative, then:
Going from step 2 to step 3 requires that , which is a separable differential equation, whose solution yields in terms of :
To verify, multiplying by 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
where is a constant.
Moving the exponential to the right-hand side, the general solution to Ordinary Differential Equation is:
In the case of a homogeneous differential equation, and the general solution to Ordinary Differential Equation is:
- .
for example, consider the differential equation
We can see that in this case
Multiplying both sides by 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 is a constant.
Solving nth order linear differential equations
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 times for an th order differential equation and combine like terms. This will yield an equation in the form
If an th order equation matches the form that is gotten after differentiating times, one can multiply all terms by the integrating factor and integrate times, dividing by the integrating factor on both sides to achieve the final result.
Example
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 , so our integrating factor is . Rearranging gives
Integrating thrice and dividing by the integrating factor yields