Integrating factors for 1st OLDE

[Square1]

So I just learned about these integrating factors and their utility in solving first order linear differential equations today. My mind is just kind of blown how it ends up simplifying the integration so much. Thats really pretty much the main thing I wanted to say... lol.

But also, how the hell are these kind of techniques discovered? Really (thats second question). Are they fluke coincidences that people just notice, "oh look multiply by e^integral of this P(x) through the expression and finding an answer to the equation is way easier!" Or is it more conscious implementation of some strategy, or noticing algebraic patterns. Anyone know about the history here? Is this all part ol Newtons package that he came up with? Sorry if the newbishness here is making some of you cringe lol. But i am just really amazed by this stuff.

Cheers

 

[HallsofIvy]

Such things are discovered in much the way many things are discovered- try this, try that, until one works! (Then publish it not telling anyone about all the mis-steps so you look really smart.)

Any First order linear differential equation can be written in the form y'+ a(x)y= f(x). An "integrating factor" is a function, [itex]\mu(x)[/itex] such that multiplying the left side of the equation makes left side a single derivative, [itex](\mu(x)y)'[/itex].

Apply the product rule to that: [itex](\mu(x)y)'= \mu y'+ \mu' y[/itex]. Of course, the point is to have that equal to [itex]\mu(y'+ a(x)y)= \mu y'+ \mu a(x)y[/itex]. Set them equal. [itex]\mu y'+ \mu' y= \mu y+ \mu a(x)y[/itex] so the [itex]\mu y'[/itex] terms cancel leaving [itex]\mu' y= \mu a(x)y[/itex] which gives the separable equation for [itex]\mu[/itex], [itex]\mu'= a(x)\mu[/itex] that is easy to solve.

Actually, any first order differential equation, linear or not, has an "integrating factor"- but it is only with linear equations that we can get a simple formula.

 

[Square1]

Haha I see I see. And what about the discovery of these integration factors? I always wonder how much of what I learn in class is the stuff that Newton or leibniz (and even some others I believe) had to work with?