Integrating Factor

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The Method of Integrating Factors

Let and be functions of and consider the following first order differential equation:

If we multiply the both sides of the equation above by the function we get that:

If we can guarantee that , then notice that by applying the product rule for differentiation that we get: and substituting and we get that which is exactly the lefthand side of the equation above. Thus we get that:

The above differential equation can be solved by integrating both sides of the equation with respect to and isolating . The question now arises on how we can find such a function .

Definition: If is a first order differential equation, then is called an Integrating Factor if for we have that .

The following proposition will give us a formula for obtaining the integrating factor for differential equations in the form .

Proposition 1: If is a differential equation, then an integrating factor of this equation is given by the formula .

  • Proof: We want to find such that . We can rewrite this equation as as and then:
  • Since we only need one integrating factor to solve differential equations in the form , we can more generally note that is an integrating factor of this differential equation.

Notice that from proposition 1 that integrating factors are not unique. In fact, there are infinitely many integrating factors. This can be see when evaluating the indefinite integral in which will result in getting where is any antiderivative if and where is a constant. We will always use the simplest integrating factor in solving differential equations of this type.

Let's now look at some examples of applying the method of integrating factors.

Example 1

Find all solutions to the differential equation .

We first notice that our differential equation is in the appropriate form where and . We compute our integrating factor as:

Thus we have that for as a constant:

Example 2

Find all solutions to the differential equation .

We first rewrite our differential equation as . We note that in this form we have and . We now find an integrating factor:

Thus we have that for as a constant:

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