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. However in some cases where depends only on or , the problem reduces to a separable first-order linear differential equation. The solutions for such cases are:

or

• Inexact differential
• Exact differential equation

• Tenenbaum, Morris; Pollard, Harry (1963). "Recognizable Exact Differential Equations". Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences. New York: Dover. pp. 80–91. ISBN 0-486-64940-7.

• A solution for an inexact differential equation from Stack Exchange
• a guide for non-partial inexact differential equations at SOS math