Integrating Factor for Product Rule

Consider this First-order ODE:

$$\frac{dy}{dx}+P(x)y=Q(x)$$

We can multiply the whole equation by an integrating factor.

$$I=e^{\int P(x)dx}$$

We can rewrite the ODE as

$$\begin{array}{rl}I\frac{dy}{dx}+IP(x)y& =IQ(x)\\ Iy& =\int IQ(x)dx\end{array}$$

Derivation

By Product Rule we have:

$$\frac{d}{dx}(Iy)=I\frac{dy}{dx}+IP(x)y$$

thus,

$$I\frac{dy}{dx}+IP(x)y=IQ(x)\Rightarrow \frac{d}{dx}(Iy)=IQ(x)$$

integrating both sides yields,

$$Iy=\int IQ(x)dx$$