13.2 Integrating Factors
shows up a lot in differential equations, because it has properties that are helpful when we differentiate it. One way in which it helps us is in solving first-order linear differential equations, which are equations of the form
This can be solved using the product rule. If we define a function , we can write by the product rule that the derivative of is
This doesn’t immediately look like our equation, but if we multiply through by , we get that
What we can do here is write that the left hand side is equal to the derivative of . This only works, however, if the derivative of is equal to . 11 1 This is because And if then And thus which is just the left-hand side of the equation. If it is, we can write that
And thus we can solve the equation by integrating.