The Integrating Factor Method

The integrating factor method is a method of solving first order differential equations:
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  
\[y=x-2+ \frac{2}{x} + \frac{A}{x}e^{-x}\]
 

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