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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?

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- Thread starter vktsn0303
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- #1

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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?

- #2

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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.

- #3

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I'm sorry about the question not matching the title. I didn't realize when posting it.

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.

Your guess sounds right. Thanks.

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