An inexact differential equation is a differential equation of the form: satisfying the condition Leonhard Euler invented the integrating factor in 1739 to solve these equations.
[1] To solve an inexact differential equation, it may be transformed into an exact differential equation by finding an integrating factor
[2] Multiplying the original equation by the integrating factor gives: For this equation to be exact,
must satisfy the condition: Expanding this condition gives: Since this is a partial differential equation, it is generally difficult.
, the problem reduces to a separable first-order linear differential equation.