What is an integrating factor exactly?

• vktsn0303
In summary, the integrating factor is a tool used in solving non-homogenous linear ODEs and non-linear ODEs by converting them to exact differentials. The idea of the integrating factor may have come from mathematicians experimenting with different functions and manipulating equations to find patterns and easier forms to work with. It is a process that requires trial and error, but can lead to a better understanding of the solutions.
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?

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.

Cruikshank said:
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.

1. What is an integrating factor?

An integrating factor is a mathematical tool used in solving certain types of differential equations. It is a function that is multiplied to both sides of a differential equation to make it easier to solve.

2. How does an integrating factor work?

When multiplied to both sides of a differential equation, the integrating factor helps to create a new equation that is easier to solve. It does this by "integrating" the coefficients of the derivative term, hence the name "integrating factor."

3. What is the purpose of using an integrating factor?

The purpose of using an integrating factor is to simplify the process of solving differential equations. It helps to convert a complex equation into a more manageable form, making it easier to find a solution.

4. What types of differential equations require an integrating factor?

Differential equations that are linear, but not in standard form, typically require the use of an integrating factor. These equations have the form y' + p(x)y = q(x), where p and q are functions of x.

5. How do you find the integrating factor for a given differential equation?

The integrating factor can be found by using a formula or by solving a first-order linear differential equation. The formula is e^(∫p(x)dx), where p(x) is the coefficient of the y term. If the differential equation is not in standard form, it can be rewritten to find the integrating factor.

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