Integrating factor

of an ordinary first-order differential equation

$$P(x,y)dx+Q(x,y)dy=0$$

A function $\mu=\mu(x,y)\not\equiv0$ with the property that

$$\mu(x,y)P(x,y)dx+\mu(x,y)Q(x,y)dy=0$$

is a differential equation with total differential. E.g., for the linear equation $y'+a(x)y=f(x)$, or $(a(x)y-f(x))dx+dy=0$, the function $\mu=\exp\int a(x)dx$ is an integrating factor. If in a domain $D$ where $P^2+Q^2\neq0$ equation

has a smooth general integral $U(x,y)=C$, then it has an infinite number of integrating factors. If $P(x,y)$ and $Q(x,y)$ have continuous partial derivatives in a domain $D$ where $P^2+Q^2\neq0$, then any particular (non-trivial) solution of the partial differential equation

$$Q\frac{\partial u}{\partial x}-P\frac{\partial u}{\partial y}+\mu\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)=0\label{2}\tag{2}$$

can be taken as integrating factor, see [1]. However, a general method for finding solutions of \eqref{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)