What is an integrating factor exactly?

[vktsn0303]

While solving non-homogenous linear ODEs we make use of the integrating factor to allow us to arrive at a solution of the unknown function. Same applies to non linear ODEs where the ODEs are converted to exact differentials.
But what I don't understand is how and why would someone have come up with the idea of the integrating factor?
How would they have seen the possibility of solving the ODE with the help of an integrating factor?

 

[Cruikshank]

Your question doesn't match the question in your title, but okay, I'll take a guess. I've wondered similar things, and one idea that occurred to me is that someone who really wanted to expand their repertoire in the early days of calculus could simply take lots of functions, find their first couple of derivatives, and then construct differential equations that they satisfy. It would be a slow process but would gradually give one a feel for the shape of some solutions.
Another point is that mathematicians are always hunting for patterns, and might come across an ODE that made them think, "oh, if only there were an x there, it would be a simple product rule!" followed by, "I wonder what would happen if I multiplied the entire equation by x?" Sometimes it is possible to "force a fit"--to contort an equation so that one piece of it looks the way you want, and then see what the rest of it looks like. Essentially, in math, it is easy to take any problem and turn it into a very similar problem with the same answer. So when one is completely lost, one can try random manipulations and see if any of the other forms look easier. If all that's obvious to you already, I apologize. That's all I've got.

 

[vktsn0303]

Your question doesn't match the question in your title, but okay, I'll take a guess. I've wondered similar things, and one idea that occurred to me is that someone who really wanted to expand their repertoire in the early days of calculus could simply take lots of functions, find their first couple of derivatives, and then construct differential equations that they satisfy. It would be a slow process but would gradually give one a feel for the shape of some solutions.
Another point is that mathematicians are always hunting for patterns, and might come across an ODE that made them think, "oh, if only there were an x there, it would be a simple product rule!" followed by, "I wonder what would happen if I multiplied the entire equation by x?" Sometimes it is possible to "force a fit"--to contort an equation so that one piece of it looks the way you want, and then see what the rest of it looks like. Essentially, in math, it is easy to take any problem and turn it into a very similar problem with the same answer. So when one is completely lost, one can try random manipulations and see if any of the other forms look easier. If all that's obvious to you already, I apologize. That's all I've got.

I'm sorry about the question not matching the title. I didn't realize when posting it.
Your guess sounds right. Thanks.