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# Separable equations example (old)

## Video transcript

I think it's reasonable to do one more separable differential equation from so let's do it derivative of Y with respect to X is equal to Y cosine of X divided by 1 plus 2y squared and they give us an initial condition that Y of 0 is equal to 1 or when X is equal to 0 Y is equal to 1 and I know we did a couple already but another way to think about separable differential equations is really all you're doing is implicit differentiation in Reverse or another way to think about it is whenever you took an implicit derivative the the end product was a separable differential equation and so you know that's just a little just so hopefully forms a little bit of a connection but anyway let's just do this we have to separate the Y's from the X's let's multiply both sides times 1 plus 2y squared we get 1 plus 2y squared times dy/dx is equal to Y cosine of X we still haven't fully separated the Y's in the X's let's divide both sides of this by Y and then let's see we get 1 over Y 1 over y plus 2y squared divided by Y that's just 2y times dy/dx is equal to cosine of X I can just multiply both sides by DX 1 over y plus 2 y times dy is equal to cosine of X DX and now we can integrate both sides let's integrate both sides so what's the integral of 1 over Y with respect to Y I know your gut reaction is the natural log of Y which is correct but there's actually a slightly broader function than that whose derivative is actually 1 over Y and that's the natural log of the absolute value of y and this is just a slightly broader function because its domain is it includes positive and negative numbers it just excludes zero while natural log of y only includes numbers larger than zero so natural log of absolute value of y is nice and it's actually true that at all points other than zero its derivative is 1 over Y so it's just a slightly broader function so that's the antiderivative of 1 over y we prove that or at least we prove that the derivative of natural log of y is 1 over y plus what's the antiderivative of 2 Y with respect to Y well it's y squared is equal to I'll do the plus C on this side what's the whose derivative is cosine of X what's sine of X sine of X and then we could add the plus C you can add that plus C there and what was our initial condition Y of 0 is equal to 1 so when X is equal to 0 Y is equal to 1 so when X is equal to 0 Y is equal to 1 so Ln of the absolute value of 1 plus 1 squared is equal to sine of 0 plus C the natural log of 1 e to the what power is one will 0 plus 1 is the sine of 0 0 is equal to C so we get C is equal to 1 so the solution to this differential equation up here is I don't even have to rewrite it we figured out C is equal to 1 so we can just scratch this out we could put a 1 if the natural log of the absolute value of y plus y squared is equal to sine of X plus 1 and actually if you were to graph this you would see that Y never actually dips below or even hits the x-axis so you could actually get rid of that absolute value function there but anyway that's just a little technicality but this is the implicit form of the solution to this differential equation that makes sense because these separable differential equations are really just implicit derivatives backwards and in general one thing that's kind of fun about differential equations but kind of not as satisfying about differential equations is it really is just a whole hodgepodge of tools to solve different types of equations there isn't just one tool or one theory that solve all differential equations there are a few that will solve a certain class of differential equations but there's not just one consistent way to solve all of them and Frank and even today there are unsolved differential equations where the only way that we know how to get solutions is using a computer in numerical again and one day I'll do videos on that and actually you'll you'll find that in most applications that's what you end up doing anyway because most differential equations you encounter in science or in with any kind of science so there's economics or physics or or engineering that they often are unsolvable because they're going to have you know they might have a second or third derivatives involved and they're going to multiply I mean there's going to be really complicated very hard to solve analytically and you actually are going to solve them numerically which which is often much easier but anyway hopefully at this point you have a pretty good sense of separable equations there's just implicit differentiation backwards and it's really nothing new our next thing we'll learn is exact differential equations and then we'll go off into more and more methods and then hopefully by the end of this playlist you'll have a nice toolkit of all of the different ways to solve at least the solvable differential equations see in the next video