Why Use an Integrating Factor in Differential Equations?

[transgalactic]

$$\frac{dp}{dt}+2tp=p+4t-2$$
$$e^{\int (2t-1)dt}=e^{t^2 -t}$$
then we do

$$\frac{d}{dt}[e^{t^2 -t}]=...$$
why??
i was told that i will get the original left side of the equation times p.
i didnt get the same resultwhat is the role of integration factor
??

 

[Cyosis]

An integrating factor is a function with which you multiply the differential equation to help facilitate the solving of the differential equation.

There was a similar question here a few days ago, where I did part of the calculation.

https://www.physicsforums.com/showthread.php?t=316907

Check post 4.