Inexact differential equation

An inexact differential equation is a differential equation of the form (see also: inexact differential)
The solution to such equations came with the invention of the integrating factor by Leonhard Euler in 1739.
Solution method.
In order to solve the equation, we need to transform it into an exact differential equation. In order to do that, we need to find an integrating factor formula_2 to multiply the equation by. We'll start with the equation itself. formula_3, so we get formula_4. We will require formula_2 to satisfy formula_6. We get
After simplifying we get
Since this is a partial differential equation, it is mostly extremely hard to solve, however in some cases we will get either formula_9 or formula_10, in which case we only need to find formula_2 with a first-order linear differential equation or a separable differential equation, and as such either
or