Integrating factor

of an ordinary first-order differential equation

A function with the property that

is a differential equation with total differential. E.g., for the linear equation , or , the function is an integrating factor. If in a domain where equation

has a smooth general integral , then it has an infinite number of integrating factors. If and have continuous partial derivatives in a domain where , then any particular (non-trivial) solution of the partial differential equation

(2)

can be taken as integrating factor, see [1]. However, a general method for finding solutions of (2) does not exist, and hence it is only in exceptional cases that one succeeds in finding an integrating factor for a concrete equation , cf. [2].

References

[1]	W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[2]	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971)