In the field of differential equations, the method of using integration factors is very universally applicable and useful. Fundamentally, the method consists of multiplying a differential equation all the way through by a factor that will ultimately simplify the form of the equation.
Practically, “It is commonly used to solve 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 friend). This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.” (Wikipedia)
To see the technique’s usefulness, start with the form: (dy/dx) + f(x)y = g(x)
Multiplying through by e^(∫f(x)dx) we have: (dy/dx)e^(∫f(x)dx) + f(x)e^(∫f(x)dx)y = g(x)e^(∫f(x)dx)
Notice that the left side can be re-written as a derivative of a product, such that: d/dx (ye^(∫f(x)dx)) = g(x)e^(∫f(x)dx)
Integrating both wides with respect to x, we have: ye^(∫f(x)) = ∫g(x)e^(∫f(x)dx)dx
Solving for y explicitly, we finally arrive at: y = (∫g(x)e^(∫f(x)dx)dx)/(e^(∫f(x))).
This concept can also be expanded to differential equations of order n:
“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 ℎ()() times, dividing by the integrating factor on both sides to achieve the final result.” (Wikipedia)
