Integrating Factor

An integrating factor is a function by which an ordinary differential equation can be multiplied in order to make it integrable. For example, a linear first-order ordinary differential equation of type

(1)

where and are given continuous functions, can be made integrable by letting be a function such that

(2)

and

(3)

Then would be the integrating factor such that multiplying by gives the expression

			(4)
			(5)

using the product rule. Integrating both sides with respect to then gives the solution

(6)