Finding the First Order Integrating Factor

The integrating factor $v(x)$ is a function such that a first-order differential equation’s solution can be expressed in terms of the integration factor and a factor $Q(x)$. It is given by

$$v=e^{\int Pdx}$$

Derivation of the Integrating Factor

Introducing $v(x)$ such that $v\frac{dy}{dx}+Pvy=\frac{d}{dx}(v\cdot y)$ brings one to

$$\frac{d}{dx}(vy)=v\frac{dy}{dx}+Pvy$$

Using the derivative product rule and cancelling terms one finds out that the equation

$$y\frac{dv}{dx}=Pvy$$

will hold if

$$lnv=\int Pdx\to v=e^{\int Pdx}$$ $$\boxed{v=e^{\int Pdx}}$$