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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 7

Lesson 6: Finding general solutions using separation of variables

# Separable equations introduction

"Separation of variables" allows us to rewrite differential equations so we obtain an equality between two integrals we can evaluate. Separable equations are the class of differential equations that can be solved using this method.

## Want to join the conversation?

• Is a further explanation of + possible, please? (I can't quite understand what's happening there with the -2 before the x and the 1/2 etc ..and the subtitles ain't helping) Thanks in advance •   You are integrating:
``⌠     -x²⎮ -x e    dx⌡``

To integrate it you use the substitution `u = -x²`, and it's differential `du = -2x dx`, which reduces the integral to
`1/2∫e^u du = 1/2 e^u`
Replacing the u substitution you get
`1/2 e^(-x²)`
• dy/dx is just a notation not a fraction,so how does multiplying by dx and canceling it make sense? •  This is just a mnemonic device, although it does simply only a tiniest bit, here's a way without separating variables:

first you move all y's on one side, then all x's on the other side
then you move dy/dx to the side where you have all y terms:

`y_terms * dy/dx = x_terms`

now integrate both sides, with respect to x:
`∫ (y_terms * dy/dx) dx = ∫ x_terms dx`

`∫ dy/dx dx = y`; but `∫ 1 dy = y` as well, therefore: `∫ (dy/dx) dx = ∫ (1) dy = y`

So you end up with turning `∫ (y_terms dy/dx) dx = ∫ x_terms dx`
into `∫ y_terms dy = ∫ x_terms dx`

Also take a look at this: https://proofwiki.org/wiki/Separation_of_Variables

In general, you are always able to solve the same problem in calculus without separating dy's and dx's, that includes differential equations as well. Although that might require more time, thinking and space on your paper..

Hope that helps.
• At why does it become e^[(-x^2)/2]? Shouldn't it be e^[-x] because i cancel the 2? Properties of powers, isn't it? • This is a common mistake that I have seen. The correct answer is indeed e^[-x^2/2].

sqrt (e^[-x^2] ) = (e^[-x^2] )^(1/2) <-- in this instance, you multiply the exponents

e^[-x^2 * 1/2] = e^[-x^2/2]

From EEweb.com

a^[n/m] = (a^[1/m] ) ^n = (a^n)^[1/m]

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• around , Sal sets y equal to the principle root of the left side, because the initial condition only gives a positive y value. But why is this necessary? Wouldn't it be just as correct to say plus or minus? • isn't -x2 always x2? So it could be simplified more.
(1 vote) • At @ how can he integrate both sides with different "d's "? (one is dx and other is Dy ) . u have to integrate both sides with same dx right ? • How can you tell if differential equations are separable? • Is there a reason that there is no video on solving a first order DE with the "Linear" method?

I'm referring particularly to this:
http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx

Is it that this method isn't as useful or is it possible to do that type of problem with another method?

Or maybe it just hasn't been added yet.
I'm just trying to organize the various methods for solving a DE.
Thanks. • At Sal first moves around the derivative operators and then adds the integral operators, which as far as I'm concerned, is an abuse of notation. I understand it "works", but given that it hasn't been formally defined I find it rather arcane. Isn't there some resource where this kind of notation is rigorously defined?, or is it used like that as a convention untill it breaks? I would like to learn some way of making it more rigorous, and more importantly, consistent, because I fear I might get confused later on when trying to use these still respectable methods. So even if it is obscure and impopular, if someone knows about some altern and more meaningful notation, I would like to know.
(1 vote) • It is definitely abuse of notation, as dy/dx isn't a fraction. But, treating it like one is extremely helpful (and at the level you're at, there really isn't a good example where treating it as a fraction fails. However, there are some caveats to this idea when you get to partial derivatives, where treating ∂y/∂x as a fraction could be pretty problematic). So, what Sal did isn't wrong from a non-rigorous point of view.

You'll need much more advanced Mathematics to be able to define differentials rigorously. So, just for now, thinking of it as a fraction will work (though yes, it really isn't, as dy and dx are infinitesimals, and they behave pretty differently as compared to finite numbers) 