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Over the last few videos we've been talking about 1st order differential equations but we haven't really had
any techniques to solve them we've been looking at approximation
techniques like slope fields and Euler's Method. Well we're finally ready
to look at our first technique that will actually solve
a differential equation and not give us an approximation
to a solution. So as long as your differential equation is first order
you can use this technique called separation of variables - that's what we're
going to take a look in this video. So, first of all
what is separable mean? Well, a first order differential equation is called
separable if you can write it like this. Normally, you
remember how you can isolate the dy/dx on the
left hand side, and then you can throw everyone else on the right hand side
and it becomes a function muddled with x's and y's. We don't
really care what they are or where we put them.
We just toss everything in on the right hand side
and it becomes a function of x and y. That's not
necessarily separable. Let me show you what separable means.
You do get terms that get x and y on the right hand side, but they have
to be placed in a very specific fashion. It has to
be able to be written as a function of x, call it p(x), times a function of y
we'll call it q(y). The letters p and q are not important. You could have called
them f and g or what have you, but the important thing is that
the right hand side is a function of x times a
function of y, and we can separate those two into a product - not into a sum, but a product.
Then, we would call this guy separable, and we have a way of solving it.
Alright, so here's the idea behind separation of variables.
If you have dy/dx equals a function of x times a function of y,
then what you're able to do is sort these guys on either side of the equals sign.
We can separate the x's to be on one side, and the y's to be on the other. Specifically,
we could write it like this. We could divide the q(y) to get it over here with the dy,
so we have q(y)dy, so we only have y's on the left. And lets take that dx and multiply him to the right to
be over here with the p(x). And so, as the name suggests, these variables have been separated. We have y's on this side
and you have x's on this side, and that's what we're after. Now, what do we do at this point? Well, you can see this is primed and ready
to do some integration. Alright, so we do next, since we only have y's on the left lets integrate this
expression with respect to y and lets integrate this expression with respect to x, and what happened
when you do that is the differentials are removed and you don't have a derivative anymore, you just have
one expression equal to another one and this will wind up being your solution to the differential equation. So, we're
using integration to get rid of the derivative here. Now, this is only possible when the 1st order differential equation
is separable like this. If you can't write it like this then you're not going to be able to
separate the variables and you won't be able to integrate. So let me show you how this works
with a short example. This is a very easy example just to get our feet wet - we'll do some more difficult
examples in the later videos. But let's say dy/dx equals 2x times y. This is obviously separable
because its 2x times y. It's a function of x times a function of y, so lets separate
these variables. We'll move the dx to the right and we'll have 2x dx, and I'll move
the y I'll divide it to the left and we'll have 1/y dy. Ok, so I've separated the variables,
let me integrate these both now. The left side would become natural log of the absoute
value of y - that's a good old fashioned Calc I integral - equals... the integral
of 2x would be x squared, plus C. Solve for the y if that's possible. It's not always
possible, but if we can let's solve for the y and in this case we can - y by raising both sides
'e' to the left hand side and 'e' to the right hand side to get rid
of the logarithm, we would get e to the x squared times e to the C, but I'll just write a generic
constant out front here. So, my point is not so much to technically work through an
example, but to illustrate the process of separation of variables. So, this guy would be the solution to this
differential equation. You can check it... Take his derivative and see if that winds up being
2x times the original. So, anyways, what I want to leave you with in this video is just a short quiz.
Now, we're not going to work these out in this video, but I want you to jot
these down and we'll work them out in the next video here. I'm giving you a sneak peak - this is
what's going to be covered in the next video. So, my question is are these three separable?
Is there some algebra you can do to write these guys once you solve for dy/dx
as a function of x times a function of y? It's not quite as easy as it sounds.
Some of these are a little tricky, so take it slow, do it carefully, and I'll give you the answers in the next video.
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