It can be used for equations of the form
\[f(x)\frac{dy}{dx}+yg(x)=h(x)\]
Divide by
\[f(x)\]
to get\[\frac{dy}{dx}+ \frac{yg(x)}{f(x)}= \frac{h(x)}{f(x)}\]
Multiply by a factor
\[e^{\int \frac{g(x)}{f(x)}dx}\]
so the equation becomes\[\frac{dy}{dx}e^{\int \frac{g(x)}{f(x)}dx}+ye^{\int \frac{g(x)}{f(x)}dx} \frac{g(x)}{f(x)}=h(x)e^{\int \frac{g(x)}{f(x)}dx}\]
We can write this as
\[\frac{d}{dx}(ye^{\int \frac{g(x)}{f(x)}dx})=h(x)e^{\int \frac{g(x)}{f(x)}dx}\]
Then
\[ye^{\int \frac{g(x)}{f(x)}dx}= \int h(x)e^{\int \frac{g(x)}{f(x)}dx} \rightarrow y= \frac{\int h(x)e^{\int \frac{g(x)}{f(x)}dx}}{e^{\int \frac{g(x)}{f(x)}dx}}\]
.Example:
\[x \frac{dy}{dx}+(x+1)y=x^2\]
,Divide by
\[x\]
.\[\frac{dy}{dx}+(1+ \frac{1}{x})y=x\]
,The integrating factor is
\[e^{\int (1+ \frac{1}{x})dx}=e^{x+lnx}=e^xe^{lnx}=e^xx=xe^x\]
.The equation becomes
\[xe^x \frac{dy}{dx}+ (1+ \frac{1}{x}) xe^x y= x^2e^x+\]
We can write this as
\[\frac{d}{dx}(yxe^x)=x^2e^x\]
Integrating both sides
\[xe^xy=x^2e^x-2xe^x+2e^x+A\]
Then
\[y=x-2+ \frac{2}{x} + \frac{A}{x}e^{-x}\]